text: 2 3. IDENTIFICATION OF HAMILTONIAN WITH ENERGY MISLEADS Anthony Leggett has surmised soundly that ”Perhaps our naive tendency to identify the Hamiltonian with the ’energy’ is (as in some cases involving time-dependent forces) misleading?” [22]. 3.1. Energy expended for the current in magnetic ﬁeld Indeed, the energy of electric current circulating in a perfect conductor deduced from the classical Hamiltonian (16.10) in [23] 1 − (cid:82) H = (p qA)2 + qφ (7) FIG. 1: Cl: The electric current I circulating in the (cid:82) 2m (cid:82) 2 loop induces magnetic dipole moment M = I S. Moment × (cid:82) m 2 τ = Mm B of force FL acts on this loop in magnetic ﬁeld turns out to be j−ust the kinetic energy V dV nqH = B. The moment of mechanical force F should be applied dV n (1/2m)(p qA)2 = dlsn mv2/2 = L I2/2 at and the energy E = π dθM B sin θ m= 2M B should be wVeak scqreening L (cid:28) L as welll as sin the quantkum case M 0 m z m z (cid:90) (cid:90) k expended in order to overturn the loop when the constant (6). This energy should be expended value of the current I is maintained with the help of power 2 source PS. Clockwise current changes into anticlockwise cur- dI (L + L)I2 ≈ L I2 rent relatively the Bz direction after this turning-over. Qu: dtIV = dtI(Lk + L) dt = k 2 2 k2 2 (8) The direction of the persistent current in superconducting t t loop changes with quantum number n change as a result of by a power source in order to create the current I in the transition of a loop segment in normal state (black) with (cid:28) 2 a loop with the inductance L L in zero magnetic a non-zero resistance R > 0 and posterior retrieval it in su- k perconducting state. ﬁeld, Fig.2Cl.(cid:90)But the pow(cid:90)er source should expend an additional energy (cid:82) (cid:82) loop with hom(cid:82)ogeneous Coop(cid:82)er pair density |Ψ|2 = n dΦ ∗ | | −s dtI V = I dt = I Φ (9) should (cid:82)equal dV Ψ HΨ = dV Ψ 2[(1/2m)(p 2 2 dt 2 V V t t qA)2 + U] = dlsn mv2 + dV n U. The potential − l s 2 V s in order to provide zero electric ﬁeld V dΦ/dt = 0 and energy dV n U does not depend on magnetic ﬂux Φ and is noVt co(cid:73)nssidered in the the(cid:73)ory of quantization. The to maintain the current I2 when the externally produced magnetic ﬁeld changes from B = 0 to B = Φ/S, Fig.2Cl. kinetic energy of Cooper pairs The energy (9) deduced from the history (involving time- Ek = dlsns m2v2 = Iqp dlm2v = LkIp2/2 (6) d−eMpemndBenotf fmoracgense)toicf dthipeosletamteomeqeunatlsMthme =enIe2rSgyinEmMag=- l l netic ﬁeld B. This energy is positive when I V > 0 and 2 d(cid:82)oes not depend on directio(cid:82)n of the velocity v or the negative when I2V < 0 in (9). c(cid:82)urrent Ip. Thus, two permitted state n and n + 1 Thus the total energy(cid:28)of the current circulating in the with diﬀerent angular momentum have the same energy loop with the section s λ2 is the sum E = E +E ∗ L tot k M V dV Ψ HΨ = LkIp2m/2 + V dV Uns = Ipm0.5Φ0/2 + of the kinetic energy (8) and the energy of magnetic dV Un at Φ = (n + 0.5)Φ and the energy diﬀerence dipole moment in magnetic ﬁeld (9). This energy can be V s 0 (cid:15) = 0 according to the canonical Hamiltonian (5). easily deduced for any case when magnetic ﬁeld B = Φ/S But we know that the energy of two states hav- appears after the current I because the current I con- 2 2 ing diﬀerent magnetic moment in non-zero magnetic stant in time has no inﬂuence on the energy expended ﬁeld should be diﬀerent. Clockwise electric current I by a power source producing the magnetic ﬁeld. It is p can be obtained from anticlockwise current I with the obvious this energy must have the same value when the (cid:82) p help of the turning-over of the loop, Fig.1Cl. It is current I appears after magnetic ﬁeld B = Φ/S. The 2 well known that we should expand the energy E = energy E is provided in this case by the power source M M π dθM B sin θ = 2M B in order the rotate the mag- producing the magnetic ﬂux due to the mutual induc- 0 m z m z netic dipole moment M = I S in magnetic ﬁeld B tance. This energy can be easily calculated in few cases, m pm z [1]. ”But when we go over to the Hamiltonian formalism for example when the mutual inductance L between 1,2 by the standard ’canonical’ procedure, the total Hamil- two loops equals the inductance L , L of each of the − 1 2 tonian (1/2m)(p qA)2 turns out to be just the kinetic loops. The equalities L = L = L take place when 1,2 1 2 energy mv2/2! Where has the ’magnetic’ energy gone?” the magnetic ﬂux inside the loop is induced with the help [22]. of a similar loop, located side by side, Fig.2Qu. ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: IDENTIFICATION

text: OF

text: HAMILTONIAN

text: WITH

text: ENERGY

text: MISLEADS

text: Anthony

text: Leggett

text: has

text: surmised

text: soundly

text: that

text: ”Perhaps

text: our

text: naive

text: tendency

text: to

text: identify

text: the

text: Hamiltonian

text: with

text: the

text: ’energy’

text: is

text: (as

text: in

text: some

text: cases

text: involving

text: time-dependent

text: forces)

text: misleading?”

text: Energy

text: expended

text: for

text: the

text: current

text: in

text: magnetic

text: ﬁeld

text: Indeed,

text: the

text: energy

text: of

text: electric

text: current

text: circulating

text: in

text: a

text: perfect

text: conductor

text: deduced

text: from

text: the

text: classical

text: Hamiltonian

text: in

text: H

text: (p

text: qφ

text: FIG.

text: Cl:

text: The

text: electric

text: current

text: I

text: circulating

text: in

text: the

text: loop

text: induces

text: magnetic

text: dipole

text: moment

text: M

text: I

text: S.

text: Moment

text: m

text: τ

text: Mm

text: B

text: of

text: force

text: FL

text: acts

text: on

text: this

text: loop

text: in

text: magnetic

text: ﬁeld

text: turns

text: out

text: to

text: be

text: j−ust

text: the

text: kinetic

text: energy

text: V

text: dV

text: nqH

text: B.

text: The

text: moment

text: of

text: mechanical

text: force

text: F

text: should

text: be

text: applied

text: dV

text: n

text: dlsn

text: L

text: at

text: and

text: the

text: energy

text: E

text: π

text: dθM

text: B

text: sin

text: θ

text: m=

text: B

text: should

text: be

text: wVeak

text: scqreening

text: L

text: L

text: as

text: welll

text: as

text: sin

text: the

text: quantkum

text: case

text: M

text: m

text: z

text: m

text: z

text: k

text: expended

text: in

text: order

text: to

text: overturn

text: the

text: loop

text: when

text: the

text: constant

text: This

text: energy

text: should

text: be

text: expended

text: value

text: of

text: the

text: current

text: I

text: is

text: maintained

text: with

text: the

text: help

text: of

text: power

text: source

text: PS.

text: Clockwise

text: current

text: changes

text: into

text: anticlockwise

text: cur-

text: dI

text: (L

text: L

text: rent

text: relatively

text: the

text: Bz

text: direction

text: after

text: this

text: turning-over.

text: Qu:

text: dtIV

text: dtI(Lk

text: L)

text: dt

text: k

text: The

text: direction

text: of

text: the

text: persistent

text: current

text: in

text: superconducting

text: t

text: t

text: loop

text: changes

text: with

text: quantum

text: number

text: n

text: change

text: as

text: a

text: result

text: of

text: by

text: a

text: power

text: source

text: in

text: order

text: to

text: create

text: the

text: current

text: I

text: in

text: the

text: transition

text: of

text: a

text: loop

text: segment

text: in

text: normal

text: state

text: (black)

text: with

text: a

text: loop

text: with

text: the

text: inductance

text: L

text: L

text: in

text: zero

text: magnetic

text: a

text: non-zero

text: resistance

text: R

text: and

text: posterior

text: retrieval

text: it

text: in

text: su-

text: k

text: perconducting

text: state.

text: ﬁeld,

text: the

text: source

text: should

text: expend

text: an

text: additional

text: energy

text: loop

text: with

text: pair

text: density

text: n

text: dΦ

text: −s

text: dtI

text: V

text: I

text: dt

text: I

text: Φ

text: should

text: dV

text: Ψ

text: HΨ

text: dV

text: Ψ

text: dt

text: V

text: V

text: t

text: t

text: U]

text: dlsn

text: dV

text: n

text: U.

text: The

text: potential

text: l

text: s

text: V

text: s

text: in

text: order

text: to

text: provide

text: zero

text: electric

text: ﬁeld

text: V

text: dΦ/dt

text: and

text: energy

text: dV

text: n

text: U

text: does

text: not

text: depend

text: on

text: magnetic

text: ﬂux

text: Φ

text: and

text: is

text: noVt

text: in

text: the

text: of

text: quantization.

text: The

text: to

text: maintain

text: the

text: current

text: when

text: the

text: externally

text: produced

text: magnetic

text: ﬁeld

text: changes

text: from

text: B

text: to

text: B

text: Φ/S,

text: kinetic

text: energy

text: of

text: Cooper

text: pairs

text: The

text: energy

text: deduced

text: from

text: the

text: history

text: (involving

text: time-

text: Ek

text: dlsns

text: Iqp

text: d−eMpemndBenotf

text: fmoracgense)toicf

text: dthipeosletamteomeqeunatlsMthme

text: l

text: l

text: netic

text: ﬁeld

text: B.

text: This

text: energy

text: is

text: positive

text: when

text: I

text: V

text: and

text: not

text: depend

text: on

text: of

text: the

text: velocity

text: v

text: or

text: the

text: negative

text: when

text: in

text: Ip.

text: Thus,

text: two

text: permitted

text: state

text: n

text: and

text: n

text: Thus

text: the

text: total

text: the

text: current

text: circulating

text: in

text: the

text: with

text: diﬀerent

text: angular

text: momentum

text: have

text: the

text: same

text: energy

text: loop

text: with

text: the

text: section

text: s

text: λ2

text: is

text: the

text: sum

text: E

text: E

text: +E

text: L

text: tot

text: k

text: M

text: V

text: dV

text: Ψ

text: HΨ

text: V

text: dV

text: Uns

text: of

text: the

text: kinetic

text: energy

text: and

text: the

text: energy

text: of

text: magnetic

text: dV

text: Un

text: at

text: Φ

text: (n

text: and

text: the

text: energy

text: diﬀerence

text: dipole

text: moment

text: in

text: magnetic

text: ﬁeld

text: This

text: energy

text: can

text: be

text: V

text: s

text: according

text: to

text: the

text: canonical

text: Hamiltonian

text: easily

text: deduced

text: for

text: any

text: case

text: when

text: magnetic

text: ﬁeld

text: B

text: Φ/S

text: But

text: we

text: know

text: that

text: the

text: energy

text: of

text: two

text: states

text: hav-

text: appears

text: after

text: the

text: current

text: I

text: because

text: the

text: current

text: I

text: con-

text: ing

text: diﬀerent

text: magnetic

text: moment

text: in

text: non-zero

text: magnetic

text: stant

text: in

text: time

text: has

text: no

text: inﬂuence

text: on

text: the

text: energy

text: expended

text: ﬁeld

text: should

text: be

text: diﬀerent.

text: Clockwise

text: electric

text: current

text: I

text: by

text: a

text: power

text: source

text: producing

text: the

text: magnetic

text: ﬁeld.

text: It

text: is

text: p

text: can

text: be

text: obtained

text: from

text: anticlockwise

text: current

text: I

text: with

text: the

text: obvious

text: this

text: energy

text: must

text: have

text: the

text: same

text: value

text: when

text: the

text: p

text: help

text: of

text: the

text: turning-over

text: of

text: the

text: loop,

text: It

text: is

text: current

text: I

text: appears

text: after

text: magnetic

text: ﬁeld

text: B

text: Φ/S.

text: The

text: well

text: known

text: that

text: we

text: should

text: expand

text: the

text: energy

text: E

text: energy

text: E

text: is

text: provided

text: in

text: this

text: case

text: by

text: the

text: power

text: source

text: M

text: M

text: π

text: dθM

text: B

text: sin

text: θ

text: B

text: in

text: order

text: the

text: rotate

text: the

text: mag-

text: producing

text: the

text: magnetic

text: ﬂux

text: due

text: to

text: the

text: mutual

text: induc-

text: m

text: z

text: m

text: z

text: netic

text: dipole

text: moment

text: M

text: I

text: S

text: in

text: magnetic

text: ﬁeld

text: B

text: tance.

text: This

text: energy

text: can

text: be

text: easily

text: calculated

text: in

text: few

text: cases,

text: m

text: pm

text: z

text: ”But

text: when

text: we

text: go

text: over

text: to

text: the

text: Hamiltonian

text: formalism

text: for

text: example

text: when

text: the

text: mutual

text: inductance

text: L

text: between

text: by

text: the

text: standard

text: ’canonical’

text: procedure,

text: the

text: total

text: Hamil-

text: two

text: loops

text: equals

text: the

text: inductance

text: L

text: L

text: of

text: each

text: of

text: the

text: tonian

text: turns

text: out

text: to

text: be

text: just

text: the

text: kinetic

text: loops.

text: The

text: equalities

text: L

text: L

text: L

text: take

text: place

text: when

text: energy

text: Where

text: has

text: the

text: ’magnetic’

text: energy

text: gone?”

text: the

text: magnetic

text: ﬂux

text: inside

text: the

text: loop

text: is

text: induced

text: with

text: the

text: help

text: of

text: a

text: similar

text: loop,

text: located

text: side

text: by

text: side,

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Prepared for submission to JHEP Non-global logarithms at ﬁnite beyond N c leading order Kamel Khelifa-Kerfaa and Yazid Delendab aDépartement de Physique, Faculté des Sciences, Université Hassiba Benbouali de Chlef, Chlef, Algeria bDépartement des Sciences de la Matière, Faculté des Sciences, Université Hadj Lakhdar, Batna, Algeria E-mail: kamel.kkhelifa@gmail.com, yazid.delenda@gmail.com Abstract: We analytically compute non-global logarithms at ﬁnite fully up to 4 loops N c − and partially at 5 loops, for the hemisphere mass distribution in e+e → di-jets to leading logarithmic accuracy. Our method of calculation relies solely on integrating the eikonal squared-amplitudes for the emission of soft energy-ordered real-virtual gluons over the ap- propriate phase space. We show that the series of non-global logarithms in the said distri- bution exhibits a pattern of exponentiation thus conﬁrming — by means of brute force — previous ﬁndings. In the large- limit, our results coincide with those recently reported N c in literature. A comparison of our proposed exponential form with all-orders numerical solutions is performed and the phenomenological impact of the ﬁnite- corrections is dis- N c cussed. Keywords: QCD Phenomenology, Jets ArXiv ePrint: 1501.00475

text: Prepared

label: paragraph

text: for

text: submission

text: to

text: JHEP

text: Non-global

label: title

text: logarithms

text: at

text: ﬁnite

text: beyond

text: N

text: c

text: leading

text: order

text: Kamel

label: author

text: Khelifa-Kerfaa

text: and

text: Yazid

text: Delendab

text: aDépartement

text: de

text: Physique,

text: Faculté

text: des

text: Sciences,

text: Université

text: Hassiba

text: Benbouali

text: de

text: Chlef,

text: Chlef,

text: Algeria

text: bDépartement

text: des

text: Sciences

text: de

text: la

text: Matière,

text: Faculté

text: des

text: Sciences,

text: Université

text: Hadj

text: Lakhdar,

text: Batna,

text: Algeria

text: E-mail:

text: kamel.kkhelifa@gmail.com,

text: yazid.delenda@gmail.com

text: Abstract:

text: We

label: abstract

text: analytically

text: compute

text: non-global

text: logarithms

text: at

text: ﬁnite

text: fully

text: up

text: to

text: loops

text: N

text: c

text: and

text: partially

text: at

text: loops,

text: for

text: the

text: hemisphere

text: mass

text: distribution

text: in

text: e+e

text: di-jets

text: to

text: leading

text: logarithmic

text: accuracy.

text: Our

text: method

text: of

text: calculation

text: relies

text: solely

text: on

text: integrating

text: the

text: eikonal

text: squared-amplitudes

text: for

text: the

text: emission

text: of

text: soft

text: energy-ordered

text: real-virtual

text: gluons

text: over

text: the

text: ap-

text: propriate

text: phase

text: space.

text: We

text: show

text: that

text: the

text: series

text: of

text: non-global

text: logarithms

text: in

text: the

text: said

text: distri-

text: bution

text: exhibits

text: a

text: pattern

text: of

text: exponentiation

text: thus

text: conﬁrming

text: by

text: means

text: of

text: brute

text: force

text: previous

text: ﬁndings.

text: In

text: the

text: large-

text: limit,

text: our

text: results

text: coincide

text: with

text: those

text: recently

text: reported

text: N

text: c

text: in

text: literature.

text: A

text: comparison

text: of

text: our

text: proposed

text: exponential

text: form

text: with

text: all-orders

text: numerical

text: solutions

text: is

text: performed

text: and

text: the

text: phenomenological

text: impact

text: of

text: the

text: ﬁnite-

text: corrections

text: is

text: dis-

text: N

text: c

text: cussed.

text: Keywords:

text: QCD

text: Phenomenology,

text: Jets

text: ArXiv

text: ePrint:


text: (cid:90) (cid:18) (cid:90) (cid:19) ΣA(ρ) = − dΦ ΘρΘin WR × dΦ(cid:0) Θρ(cid:1)Θin WR × dΦ ΘρΘin WR 3 1 1 1 1 2 2 2 2 3 3 3 3 x1>x2>x3 3 = 1 − dΦ ΘρΘin WR = 1 ΣP 3 = ΣP(ρ) , (3.4) 3! 1 1 1 1 3! 1 3 which is simply the expansion of the Sudakov ΣP(ρ) at this order. The factor 1/3! accounts for the fact that the integrand in eq. (3.4) is completely symmetric under the exchange of gluons, which means that the condition can be relaxed and x > x > x 1 2 3 the result multiplied by . The integral is then factored out into the product of three 1/3! identical integrals, each of them resembling the one-loop result eq. (2.15). The sec(cid:90)ond contribution to Σ (ρ) is: (cid:90) 3 (cid:90) ΣB(ρ) = dΠ Θin WRΘoutΘin WRR + dΠ Θin WRΘoutΘin WRR+ 3 123 1 1 2 3 23 123 2 2 1 3 13 x1>x2>x3 x1>x2>x3 + dΠ Θin WRΘoutΘin WRR . (3.5) 123 3 3 1 2 12 x1>x2>x3 By swapping in the second integral of eq. (3.5), and performing the successive k ↔ k 1 2 permutations: k ↔ k then k ↔ k in the third integral of the same equation, ΣB(ρ) 1 2 1 3 3 becomes: (cid:90) (cid:90) (cid:90) ΣB(ρ) = dΠ Θin WRΘoutΘin WRR + dΠ Θin WRΘoutΘin WRR+ 3 123 1 1 2 3 23 123 1 1 2 3 23 x1>x2>x3 x2>x1>x3 + dΠ Θin WRΘoutΘin WRR . (3.6) 123 1 1 2 3 23 x2>x3>x1 The three integrands in eq. (3.6) are identical except for the region of integration over transverse momenta fractions. Thus we unify them into a single integral with the region of integration expressed by: Θ(x − x )Θ(x − x ) + Θ(x − x )Θ(x − x ) + Θ(x − x )Θ(x − x ) = Θ(x − x ) . 1 2 2 3 2 1 1 3 2 3 3 1 2 3 (3.7) Hence we write: (cid:18) (cid:90) (cid:19) (cid:18) (cid:90) (cid:19) ΣB(ρ) = − dΦ ΘρΘin WR × − dΠ ΘoutΘinWRR 3 1 1 1 1 23 2 3 23 x2>x3 =ΣP(ρ) × ΣNG(ρ) . (3.8) 1 2 – 13 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: ΣA(ρ)

text: dΦ

text: ΘρΘin

text: WR

text: WR

text: dΦ

text: ΘρΘin

text: WR

text: dΦ

text: ΘρΘin

text: WR

text: ΣP

text: ΣP(ρ)

text: which

text: is

text: simply

text: the

text: expansion

text: of

text: the

text: Sudakov

text: ΣP(ρ)

text: at

text: this

text: order.

text: The

text: factor

text: accounts

text: for

text: the

text: fact

text: that

text: the

text: integrand

text: in

text: eq.

text: is

text: completely

text: symmetric

text: under

text: the

text: exchange

text: of

text: gluons,

text: which

text: means

text: that

text: the

text: condition

text: can

text: be

text: relaxed

text: and

text: x

text: x

text: x

text: the

text: result

text: multiplied

text: by

text: The

text: integral

text: is

text: then

text: factored

text: out

text: into

text: the

text: product

text: of

text: three

text: identical

text: integrals,

text: each

text: of

text: them

text: resembling

text: the

text: one-loop

text: result

text: eq.

text: The

text: contribution

text: to

text: Σ

text: (ρ)

text: is:

text: ΣB(ρ)

text: dΠ

text: Θin

text: WRΘoutΘin

text: WRR

text: dΠ

text: Θin

text: WRΘoutΘin

text: WRR+

text: dΠ

text: Θin

text: WRΘoutΘin

text: WRR

text: By

text: swapping

text: in

text: the

text: second

text: integral

text: of

text: eq.

text: and

text: performing

text: the

text: successive

text: k

text: k

text: permutations:

text: k

text: k

text: then

text: k

text: k

text: in

text: the

text: third

text: integral

text: of

text: the

text: same

text: equation,

text: ΣB(ρ)

text: becomes:

text: ΣB(ρ)

text: dΠ

text: Θin

text: WRΘoutΘin

text: WRR

text: dΠ

text: Θin

text: WRΘoutΘin

text: WRR+

text: dΠ

text: Θin

text: WRΘoutΘin

text: WRR

text: The

text: three

text: integrands

text: in

text: eq.

text: are

text: identical

text: except

text: for

text: the

text: region

text: of

text: integration

text: over

text: transverse

text: momenta

text: fractions.

text: Thus

text: we

text: unify

text: them

text: into

text: a

text: single

text: integral

text: with

text: the

text: region

text: of

text: integration

text: expressed

text: by:

text: Θ(x

text: x

text: )Θ(x

text: x

text: Θ(x

text: x

text: )Θ(x

text: x

text: Θ(x

text: x

text: )Θ(x

text: x

text: Θ(x

text: x

text: Hence

text: we

text: write:

text: ΣB(ρ)

text: dΦ

text: ΘρΘin

text: WR

text: dΠ

text: ΘoutΘinWRR

text: =ΣP(ρ)

text: ΣNG(ρ)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:101) where Uˆ is the sum of all terms which when operate on the squared amplitudes WX 4 1234 and is summed over give zero. Acting by the measurement operator on the various X sq(cid:88)uared amplitudes and summing(cid:0)over conﬁgurations we obtain: Uˆ WX = −ΘρΘρΘρΘρΘin WVVVR + Θout WRVVR + Θout WVRVR+ 4 1234 1 2 3 4 4 1234 1 1234 2 1234 (cid:1) X +Θout WVVRR + ΘoutΘout WRRVR + ΘoutΘout WVRRR+ 3 1234 1 2 1234 2 3 1234 +ΘoutΘout WRVRR + ΘoutΘoutΘout WRRRR . (3.15) 1 3 1234 1 2 3 1234 From eqs. (3.2) and (3.15), it should be clear how the result of the action of the measurement operator on the squared amplitudes at mth loop order would look like: the softest gluon is always inside , • H R • each real gluon k is associated with a step function Θout , i i • virtual gluons are associated with neither Θin nor Θout (i.e., they can either be in or out of ). H R The hemisphere mass distribution at fourth order may then be cast in the form (2.5). Substituting the various matrix-elements squared we can split the hemisphere mass distri- bution,(cid:90)Σ (ρ), into ﬁve parts: Σ = ΣA + ΣB + ΣC + ΣD + ΣE, with: 4 4 4 4 4 4 4 (cid:90) (cid:16) ΣA = dΠ ΘinΘinΘinΘin WRWRWRWR , 4 1234 1 2 3 4 1 2 3 4 x1>x2>x3>x4 (cid:17) ΣB = − dΠ ΘoutΘinΘinΘin WRRWRWR + 2 ↔ 3 + 2 ↔ 4+ 4 1234 1 2 3 4 12 3 4 (cid:90) x1>x2>x3>x4 (cid:110) (cid:16) (cid:17) and then then + [1 ↔ 3 2 ↔ 4] + [2 ↔ 1 1 ↔ 4] + [2 ↔ 1 1 ↔ 3] , (cid:111) ΣC = dΠ ΘoutΘinΘin WR WRVR + Θout WRRR + 3 ↔ 4+ 4 1234 1 3 4 4 123 2 123 (cid:90)x1>x2>x3>x4 (cid:16) (cid:17) then then then + [3 ↔ 2 2 ↔ 4] + [1 ↔ 2 1 ↔ 3 1 ↔ 4] , (cid:90) ΣD = dΠ ΘoutΘinΘoutΘin WRRWRR + 2 ↔ 3 + 2 ↔ 3 then 3 ↔ 4 , 4 1234 1 2 3 4 12 34 x1>x2>x3>x4 (cid:16) (cid:17) ΣE = − dΠ ΘoutΘin× 4 1234 1 4 x1>x2>x3>x4 × WRVVR + Θout WRVRR + Θout WRRVR + ΘoutΘout WRRRR , (3.16) 1234 3 1234 2 1234 2 3 1234 where terms in the last line are the four-loop irreducible components of the squared amplitudes for the corresponding gluon conﬁgurations. The parts ΣA, ΣB, ΣC, and ΣD 4 4 4 4 completely reduce to integrals we calculated at previous orders, while the remaining ΣE part 4 is the new NGLs contribution. Let us evaluate each of these integrals separately starting with the reducible parts. – 16 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: Uˆ

text: is

text: the

text: sum

text: of

text: all

text: terms

text: which

text: when

text: operate

text: on

text: the

text: squared

text: amplitudes

text: WX

text: and

text: is

text: summed

text: over

text: give

text: zero.

text: Acting

text: by

text: the

text: measurement

text: operator

text: on

text: the

text: various

text: X

text: amplitudes

text: and

text: conﬁgurations

text: we

text: obtain:

text: Uˆ

text: WX

text: −ΘρΘρΘρΘρΘin

text: WVVVR

text: Θout

text: WRVVR

text: Θout

text: WVRVR+

text: X

text: +Θout

text: WVVRR

text: ΘoutΘout

text: WRRVR

text: ΘoutΘout

text: WVRRR+

text: +ΘoutΘout

text: WRVRR

text: ΘoutΘoutΘout

text: WRRRR

text: From

text: eqs.

text: and

text: it

text: should

text: be

text: clear

text: how

text: the

text: result

text: of

text: the

text: action

text: of

text: the

text: measurement

text: operator

text: on

text: the

text: squared

text: amplitudes

text: at

text: mth

text: loop

text: order

text: would

text: look

text: like:

text: the

text: softest

text: gluon

text: is

text: always

text: inside

text: H

text: R

text: each

text: real

text: gluon

text: k

text: is

text: associated

text: with

text: a

text: step

text: function

text: Θout

text: i

text: i

text: virtual

text: gluons

text: are

text: associated

text: with

text: neither

text: Θin

text: nor

text: Θout

text: (i.e.,

text: they

text: can

text: either

text: be

text: in

text: or

text: out

text: of

text: H

text: R

text: The

text: hemisphere

text: mass

text: distribution

text: at

text: fourth

text: order

text: may

text: then

text: be

text: cast

text: in

text: the

text: form

text: Substituting

text: the

text: various

text: matrix-elements

text: squared

text: we

text: can

text: split

text: the

text: hemisphere

text: mass

text: distri-

text: (ρ),

text: into

text: ﬁve

text: parts:

text: Σ

text: ΣA

text: ΣB

text: ΣC

text: ΣD

text: ΣE,

text: with:

text: ΣA

text: dΠ

text: ΘinΘinΘinΘin

text: WRWRWRWR

text: ΣB

text: dΠ

text: ΘoutΘinΘinΘin

text: WRRWRWR

text: and

text: then

text: then

text: ΣC

text: dΠ

text: ΘoutΘinΘin

text: WR

text: WRVR

text: Θout

text: WRRR

text: then

text: then

text: then

text: ΣD

text: dΠ

text: ΘoutΘinΘoutΘin

text: WRRWRR

text: then

text: ΣE

text: dΠ

text: ΘoutΘin×

text: WRVVR

text: Θout

text: WRVRR

text: Θout

text: WRRVR

text: ΘoutΘout

text: WRRRR

text: where

text: terms

text: in

text: the

text: last

text: line

text: are

text: the

text: four-loop

text: irreducible

text: components

text: of

text: the

text: squared

text: amplitudes

text: for

text: the

text: corresponding

text: gluon

text: conﬁgurations.

text: The

text: parts

text: ΣA,

text: ΣB,

text: ΣC,

text: and

text: ΣD

text: completely

text: reduce

text: to

text: integrals

text: we

text: calculated

text: at

text: previous

text: orders,

text: while

text: the

text: remaining

text: ΣE

text: part

text: is

text: the

text: new

text: NGLs

text: contribution.

text: Let

text: us

text: evaluate

text: each

text: of

text: these

text: integrals

text: separately

text: starting

text: with

text: the

text: reducible

text: parts.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: )t(GNΣ large N c 1 NG Σ DS 0.8 2-loops 3-loops 4-loops 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 t Figure 5. Plot of the NGLs function ΣNG(t) at large N for the range 0 ≤ t ≤ 0.14. c results were shown to be relatively straightforward to obtain, and were found to be directly related to the Riemann-Zeta function. We noticed that, up to this loop order, ﬁnite- N c corrections are absent. This was a direct consequence of the relatively simple structure of the eikonal amplitudes (up to this order) as well as the combination of real/virtual amplitudes induced by the phase space measurement operator. Within the same eikonal framework we computed the four-loops contribution to NGLs distribution. The latter turns out to be much harder than the previous two orders and the simple result in terms of a product of a single colour factor and a Zeta function breaks down. This failure originates from a break in the simple structure of the corresponding real-virtual eikonal amplitudes at four-loops order, a phenomenon that was noticed more than two decays ago [32]. Nevertheless, we were able to overcome this complexity, compute NGLs and even spot a new pattern for NGLs at and beyond this order. This pattern helped us to successfully write down the ﬁve-loops contribution to NGLs up to constant coeﬃcients which we extracted from comparisons to previous large- results [30]. We hope to be able N c to fully compute these constants in the near future. The ﬁve-loops calculation reveals that the NGLs distribution seems to exhibit a pattern of exponentiation. To this end the said distribution was cast in an exponential form with full NGLs coeﬃcients and colour factors up to four-loops in the exponent. Comparisons to large- results obtained by other authors [30] conﬁrmed our ﬁndings, N c at least in the latter limit. We then took the step forth and compared our exponential function to the all-orders numerical resummed result reported by Dasgupta and Salam [24]. To our surprise, the shape of the all-orders result was best represented by the two- loops approximation for a wide range of the evolution parameter . In the region of small t , however, adding more terms in the exponent of our resummed result yielded better t agreement, then the two-loops result, with the all-orders numerical result. This suggests – 28 – ##LTFigure##

text: )t(GNΣ

font: default

text: large

text: N

text: c

text: NG

text: Σ

text: DS

text: t

text: Figure

text: Plot

label: caption

text: of

text: the

text: NGLs

text: function

text: ΣNG(t)

text: at

text: large

text: N

text: for

text: the

text: range

text: t

text: c

text: results

text: were

text: shown

text: to

text: be

text: relatively

text: straightforward

text: to

text: obtain,

text: and

text: were

text: found

text: to

text: be

text: directly

text: related

text: to

text: the

text: Riemann-Zeta

text: function.

text: We

text: noticed

text: that,

text: up

text: to

text: this

text: loop

text: order,

text: ﬁnite-

text: N

text: c

text: corrections

text: are

text: absent.

text: This

text: was

text: a

text: direct

text: consequence

text: of

text: the

text: relatively

text: simple

text: structure

text: of

text: the

text: eikonal

text: amplitudes

text: (up

text: to

text: this

text: order)

text: as

text: well

text: as

text: the

text: combination

text: of

text: real/virtual

text: amplitudes

text: induced

text: by

text: the

text: phase

text: space

text: measurement

text: operator.

text: Within

text: the

text: same

text: eikonal

text: framework

text: we

text: computed

text: the

text: four-loops

text: contribution

text: to

text: NGLs

text: distribution.

text: The

text: latter

text: turns

text: out

text: to

text: be

text: much

text: harder

text: than

text: the

text: previous

text: two

text: orders

text: and

text: the

text: simple

text: result

text: in

text: terms

text: of

text: a

text: product

text: of

text: a

text: single

text: colour

text: factor

text: and

text: a

text: Zeta

text: function

text: breaks

text: down.

text: This

text: failure

text: originates

text: from

text: a

text: break

text: in

text: the

text: simple

text: structure

text: of

text: the

text: corresponding

text: real-virtual

text: eikonal

text: amplitudes

text: at

text: four-loops

text: order,

text: a

text: phenomenon

text: that

text: was

text: noticed

text: more

text: than

text: two

text: decays

text: ago

text: Nevertheless,

text: we

text: were

text: able

text: to

text: overcome

text: this

text: complexity,

text: compute

text: NGLs

text: and

text: even

text: spot

text: a

text: new

text: pattern

text: for

text: NGLs

text: at

text: and

text: beyond

text: this

text: order.

text: This

text: pattern

text: helped

text: us

text: to

text: successfully

text: write

text: down

text: the

text: ﬁve-loops

text: contribution

text: to

text: NGLs

text: up

text: to

text: constant

text: coeﬃcients

text: which

text: we

text: extracted

text: from

text: comparisons

text: to

text: previous

text: large-

text: results

text: We

text: hope

text: to

text: be

text: able

text: N

text: c

text: to

text: fully

text: compute

text: these

text: constants

text: in

text: the

text: near

text: future.

text: The

text: ﬁve-loops

text: calculation

text: reveals

text: that

text: the

text: NGLs

text: distribution

text: seems

text: to

text: exhibit

text: a

text: pattern

text: of

text: exponentiation.

text: To

text: this

text: end

text: the

text: said

text: distribution

text: was

text: cast

text: in

text: an

text: exponential

text: form

text: with

text: full

text: NGLs

text: coeﬃcients

text: and

text: colour

text: factors

text: up

text: to

text: four-loops

text: in

text: the

text: exponent.

text: Comparisons

text: to

text: large-

text: results

text: obtained

text: by

text: other

text: authors

text: conﬁrmed

text: our

text: ﬁndings,

text: N

text: c

text: at

text: least

text: in

text: the

text: latter

text: limit.

text: We

text: then

text: took

text: the

text: step

text: forth

text: and

text: compared

text: our

text: exponential

text: function

text: to

text: the

text: all-orders

text: numerical

text: resummed

text: result

text: reported

text: by

text: Dasgupta

text: and

text: Salam

text: To

text: our

text: surprise,

text: the

text: shape

text: of

text: the

text: all-orders

text: result

text: was

text: best

text: represented

text: by

text: the

text: two-

text: loops

text: approximation

text: for

text: a

text: wide

text: range

text: of

text: the

text: evolution

text: parameter

text: In

text: the

text: region

text: of

text: small

text: t

text: however,

text: adding

text: more

text: terms

text: in

text: the

text: exponent

text: of

text: our

text: resummed

text: result

text: yielded

text: better

text: t

text: agreement,

text: then

text: the

text: two-loops

text: result,

text: with

text: the

text: all-orders

text: numerical

text: result.

text: This

text: suggests

text: ##LTFigure##


text: [34] S. Catani and M. Grazzini, Infrared factorization of tree level QCD amplitudes at the next-to-next-to-leading order and beyond, Nucl. Phys. B570 (2000) 287–325, [Erratum ibid. B 510 (1998) 503] [hep-ph/9908523]. [35] S. Catani and M. Seymour, A general algorithm for calculating jet cross-sections in NLO QCD, Nucl. Phys. B485 (1997) 291–419, [hep-ph/9605323]. [36] S. Catani and M. Seymour, The dipole formalism for the calculation of QCD jet cross-sections at next-to-leading order, Phys. Lett. B378 (1996) 287–301, [hep-ph/9602277]. [37] M. Sjödahl, ColorMath - A package for color summed calculations in SU(N ), Eur. Phys. J. c C73 (2013), no. 2 2310, [arXiv:1211.2099]. [38] M. Sjödahl and S. Keppeler, Tools for calculations in color space, PoS DIS2013 (2013) 166, [arXiv:1307.1319]. [39] Y. Delenda and K. Khelifa-Kerfa, “Eikonal gluon bremsstrahlung at ﬁnite N to any loop c order.” In preparation. [40] R. Kelley, M. D. Schwartz, R. M. Schabinger, and H. X. Zhu, The two-loop hemisphere soft function, Phys. Rev. D84 (2011) 045022, [arXiv:1105.3676]. [41] A. Hornig, C. Lee, I. W. Stewart, J. R. Walsh, and S. Zuberi, Non-global structure of the O(α2) dijet soft function, JHEP 1108 (2011) 054, [arXiv:1105.4628]. s − [42] A. H. Hoang and S. Kluth, Hemisphere soft function at O(α2) for dijet production in e+e s annihilation, arXiv:0806.3852. [43] K. Khelifa-Kerfa, Non-global logs and clustering impact on jet mass with a jet veto distribution, JHEP 1202 (2012) 072, [arXiv:1111.2016]. [44] G. Marchesini and B. Webber, Simulation of QCD jets including soft gluon interference, Nucl. Phys. B238 (1984) 1. [45] T. Hahn, CUBA - a library for multidimensional numerical integration, Comput. Phys. Commun. 168 (2005) 78–95, [hep-ph/0404043]. [46] Y. Hatta and T. Ueda. Private communication. [47] Y. Delenda and K. Khelifa-Kerfa, On the resummation of clustering logarithms for non-global observables, JHEP 1209 (2012) 109, [arXiv:1207.4528]. [48] Y. L. Dokshitzer, G. Leder, S. Moretti, and B. Webber, Better jet clustering algorithms, JHEP 9708 (1997) 001, [hep-ph/9707323]. [49] M. Wobisch and T. Wengler, Hadronization corrections to jet cross-sections in deep inelastic scattering, hep-ph/9907280. [50] R. B. Appleby and M. H. Seymour, Nonglobal logarithms in interjet energy ﬂow with kt clustering requirement, JHEP 0212 (2002) 063, [hep-ph/0211426]. [51] A. Banﬁ and M. Dasgupta, Problems in resumming interjet energy ﬂows with k clustering, t Phys.Lett. B628 (2005) 49–56, [hep-ph/0508159]. [52] Y. Delenda, R. Appleby, M. Dasgupta, and A. Banﬁ, On QCD resummation with k t clustering, JHEP 0612 (2006) 044, [hep-ph/0610242]. [53] A. Banﬁ, M. Dasgupta, K. Khelifa-Kerfa, and S. Marzani, Non-global logarithms and jet algorithms in high-pT jet shapes, JHEP 1008 (2010) 064, [arXiv:1004.3483]. – 34 –

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text: 3.4 The Asymptotic Distribution of γ˜ In this section we formal(cid:32)ise the discussion in Section as Theorem 2. T(cid:33)o th(cid:32)is end, deﬁne (cid:33) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) √ (cid:3) ΣΠε = E(S−1Π(cid:48)εε(cid:48)ΠS−1) = EEDZ(cid:48)ε(cid:48)εεε(cid:48)Z(cid:48)Z//((√NNTT)) E ZE(cid:48)εDε(cid:48)(cid:48)Dεε/(cid:48)D( /NT T ) = ΣΣ1(cid:48) ,N ΣΣ2,N . 2,N 3,N   and note that (cid:88) (cid:88) (cid:88) (cid:2) (cid:3) (cid:88) (cid:88) (cid:104) (cid:105)   N N N N T E 1 (cid:48) (cid:48) 1 E (cid:48) (cid:48) 1 E (cid:48) Σ = Z ε ε Z = Z ε ε Z = ε2 z z , 1,N NT i i j j NT i i i i NT i,t i,t i(cid:80),t (cid:2) (cid:3) i=1 j=1 i=1 i=1 t=1 (cid:104) (cid:105) (cid:80) (cid:80) (cid:80) (cid:80) E (cid:48) (cid:48) where the second and third equality both follow from Assumption 1. Likewise, Σ = 1 N d ε ε d = 3,N T i=1 i i i i 1 N T E ε2 d (cid:80)d(cid:48) = d(cid:2)iag( 1 (cid:3)T E[ε2 ](cid:80), ..., 1(cid:80) T E(cid:104)[ε2 ]), wh(cid:105)ere d(cid:48) is the ith T × N T i=1 t=1 i,t i,t i,t × T t=1 1,t T t=1 N,t i block of D, and di,t is a N 1 zero vector (cid:80)with the ith entry replaced by 1. In the same manner, Σ = √1 N E z ε ε(cid:48)d(cid:48) = √1 N T E ε2 z d(cid:48) . In words, Σ is a p × N mat2r,iNx with NitTs ithi=c1olumin ibei ini g √1 NT T i=E1[z tε=21 ]. Fiin,talil,yt , im,t otivated by the2,aNbove, (cid:32) (cid:33)  (cid:80) NT(cid:80) t=1 i,t i,t (cid:80) (cid:80)  deﬁne the feasible sample counterpart of Σ as Πε  (cid:80) (cid:80) (cid:80) (cid:80)  ΣˆΠε = ΣΣˆˆ1(cid:48) ,N ΣΣˆˆ2,N := √N11T Ni=N1 Tt=T1 εˆεˆ2i,2tzdi,tzzi(cid:48),(cid:48)t √N11 T NNi=1 T Tt=εˆ12εˆ2id,tzid,t(cid:48)d(cid:48)i,t , 2,N 3,N NT i=1 t=1 i,t i,t i,t T i=1 t=1 i,t i,t i,t − (cid:48) − where εˆ := y z αˆ ηˆ . One could also consider constructing εˆ based on the desparsiﬁed i,t i,t i,t i i,t estimates. However, this would require running the nodewise regressions for all variables and not only those pertaining to the coeﬃcients in the hypothesis being tested resulting in a much more computationally demanding procedure. The following assumptions are needed to establish the validity of asymptotically gaussian inference of our procedure. ∨ ∨ Assumption 5. Let p˜ := p N T and assume (cid:104) (cid:105) (a) − (h1 ∨ h21{h1 (cid:54)= 0})2G¯2 (logNp)3 ϑ (log p˜)7 (log(N ∨ T ))31{h2 (cid:54)= 0} = o(1), = o(1). N T (cid:20) (cid:21) (cid:16) (cid:104) (cid:105) (cid:17) (cid:16) (cid:17) (b) − − h2G¯2 (log p)3 ϑ ∨ Nh2 s ∨ E (log(p∨N))3 ν/2 (log p˜)5 1 N 2 1 T = o(1). NT (cid:104)(cid:16) (cid:104) (cid:105) (cid:17) (cid:105) (cid:104) (cid:16) (cid:17) (cid:105) (c) − − (h ∨ h ) G¯ (log p)3 ϑ/2 ∨ (log p˜)2 1{h (cid:54)= 0} ∨ 1{h (cid:54)= 0} s2 ∨ E2 (log(p∨N))3 ν (log p˜)4 1 2 N 1 2 1 T = o(1). N (d) mineval(Σ ) is uniformly bounded away from zero and maxeval(Σ ) is uniformly bounded Πε 1,N from above. 13 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: section

text: The

text: Asymptotic

text: Distribution

text: of

text: γ˜

text: In

text: this

text: section

text: we

text: the

text: discussion

text: in

text: Section

text: as

text: Theorem

text: end,

text: deﬁne

text: ΣΠε

text: E

text: /NT

text: T

text: ,N

text: and

text: note

text: that

text: N

text: N

text: N

text: N

text: T

text: E

text: E

text: E

text: Σ

text: Z

text: ε

text: ε

text: Z

text: Z

text: ε

text: ε

text: Z

text: ε2

text: z

text: z

text: NT

text: i

text: i

text: j

text: j

text: NT

text: i

text: i

text: i

text: i

text: NT

text: i,t

text: i,t

text: E

text: where

text: the

text: second

text: and

text: third

text: equality

text: both

text: follow

text: from

text: Assumption

text: Likewise,

text: Σ

text: N

text: d

text: ε

text: ε

text: d

text: T

text: i

text: i

text: i

text: i

text: N

text: T

text: E

text: ε2

text: d

text: T

text: is

text: the

text: ith

text: T

text: N

text: T

text: i,t

text: i,t

text: i,t

text: T

text: T

text: N,t

text: i

text: block

text: of

text: D,

text: and

text: di,t

text: is

text: a

text: N

text: zero

text: vector

text: the

text: ith

text: entry

text: replaced

text: by

text: In

text: the

text: same

text: manner,

text: Σ

text: √1

text: N

text: E

text: z

text: ε

text: √1

text: N

text: T

text: E

text: ε2

text: z

text: In

text: words,

text: Σ

text: is

text: a

text: p

text: N

text: with

text: NitTs

text: ibei

text: ini

text: g

text: √1

text: NT

text: T

text: Fiin,talil,yt

text: im,t

text: otivated

text: by

text: i,t

text: i,t

text: deﬁne

text: the

text: feasible

text: sample

text: counterpart

text: of

text: Σ

text: as

text: Πε

text: ΣˆΠε

text: ,N

text: ΣΣˆˆ2,N

text: T

text: T

text: NT

text: i,t

text: i,t

text: i,t

text: T

text: i,t

text: i,t

text: i,t

text: where

text: εˆ

text: y

text: z

text: αˆ

text: ηˆ

text: One

text: could

text: also

text: consider

text: constructing

text: εˆ

text: based

text: on

text: the

text: desparsiﬁed

text: i,t

text: i,t

text: i,t

text: i

text: i,t

text: estimates.

text: However,

text: this

text: would

text: require

text: running

text: the

text: nodewise

text: regressions

text: for

text: all

text: variables

text: and

text: not

text: only

text: those

text: pertaining

text: to

text: the

text: coeﬃcients

text: in

text: the

text: hypothesis

text: being

text: tested

text: resulting

text: in

text: a

text: much

text: more

text: computationally

text: demanding

text: procedure.

text: The

text: following

text: assumptions

text: are

text: needed

text: to

text: establish

text: the

text: validity

text: of

text: asymptotically

text: gaussian

text: inference

text: of

text: our

text: procedure.

text: Assumption

text: Let

text: p˜

text: p

text: N

text: T

text: and

text: assume

text: (a)

text: ϑ

text: (log

text: (log(N

text: T

text: N

text: T

text: (b)

text: (log

text: ϑ

text: s

text: E

text: (log

text: N

text: T

text: NT

text: (c)

text: (h

text: h

text: G¯

text: (log

text: (log

text: ν

text: (log

text: N

text: T

text: N

text: (d)

text: mineval(Σ

text: is

text: uniformly

text: bounded

text: away

text: from

text: zero

text: and

text: maxeval(Σ

text: is

text: uniformly

text: bounded

text: Πε

text: from

text: above.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: RMSE Coverage Length α η γ γ γ γ γ γ Size Power 7 87 167 7 87 167 LS 4(a) DL 2.145 14.002 0.924 0.891 0.920 0.261 0.262 0.263 0.123 0.988 Ora 0.351 13.570 0.936 0.934 0.930 0.282 0.283 0.285 0.065 0.999 LS 4(b) DL 2.145 14.073 0.933 0.904 0.911 0.275 0.260 0.263 0.114 0.979 Ora 0.368 13.469 0.926 0.931 0.940 0.283 0.281 0.282 0.076 1.000 LS 4(c) DL 4.305 42.303 0.917 0.899 0.903 0.456 0.386 0.390 0.139 0.825 Ora 0.782 41.269 0.926 0.934 0.926 0.465 0.428 0.435 0.087 0.870 Table 4: Experiment 4. LS, DL and Ora: least squares including all variables, desparsiﬁed Lasso and least squares oracle. RMSE: root mean square error. Coverage: the coverage rate of the asymptotic 95% conﬁdence intervals. Length: the average length of the asymptotic 95% conﬁdence intervals. Size: size of the correct hypothesis H : (γ , γ , γ ) = (0, 0, 0). Power: the probability to reject the false 0 7 87 167 H : (γ , γ , γ ) = (0.4, 0, 0). 0 7 87 167 estimated. Thus, it is not surprising that the estimation error for α goes down while the one for η increases. The coverage rates of the oracle as well as the desparsiﬁed Lasso improve compared to Table 2. However, the length of the conﬁdence bands does not decrease as much as when T was increased in the previous experiment. The size of the signiﬁcance test decreases when increasing N while power is close to 1. Adding heteroskedasticity has no consequences while the presence of heavy tails has the usual eﬀect. RMSE Coverage Length α η γ γ γ γ γ γ Size Power 7 74 141 7 74 141 LS 5(a) DL 3.342 2.463 0.922 0.903 0.912 0.209 0.209 0.208 0.124 0.997 Ora 0.546 3.305 0.956 0.942 0.949 0.187 0.187 0.187 0.062 1.000 LS 5(b) DL 3.327 2.432 0.913 0.916 0.896 0.218 0.207 0.208 0.127 0.998 Ora 0.556 3.261 0.942 0.951 0.921 0.190 0.186 0.186 0.065 1.000 LS 5(c) DL 7.294 7.273 0.936 0.910 0.916 0.363 0.307 0.304 0.101 0.920 Ora 1.072 9.693 0.952 0.936 0.951 0.326 0.297 0.293 0.061 0.955 Table 5: Experiment 5. LS, DL and Ora: least squares including all variables, desparsiﬁed Lasso and least squares oracle. RMSE: root mean square error. Coverage: the coverage rate of the asymptotic 95% conﬁdence intervals. Length: the average length of the asymptotic 95% conﬁdence intervals. Size: size of the correct hypothesis H : (γ , γ , γ ) = (0, 0, 0). Power: the probability to reject the false 0 7 74 141 H : (γ , γ , γ ) = (0.4, 0, 0). 0 7 74 141 Table 5 contains a setting with more than 1000 variables. The main message of the previous tables prevails even in this setting: the coverage of the Lasso-based conﬁdence intervals is almost as good as the ones based on the oracle. On the other hand, the bands of the former are now slightly wider than the ones of the latter. Both procedures have power close to one while the Lasso-based test is a bit oversized compared to the oracle based test. Heteroskedasticity 21 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: RMSE

label: table

text: Coverage

text: Length

text: α

text: η

text: γ

text: γ

text: γ

text: γ

text: γ

text: γ

text: Size

text: Power

text: LS

text: DL

text: Ora

text: LS

text: DL

text: Ora

text: LS

text: DL

text: Ora

text: Table

text: Experiment

text: LS,

text: DL

text: and

text: Ora:

text: least

text: squares

text: including

text: all

text: variables,

text: desparsiﬁed

text: Lasso

text: and

text: least

text: squares

text: oracle.

text: RMSE:

text: root

text: mean

text: square

text: error.

text: Coverage:

text: the

text: coverage

text: rate

text: of

text: the

text: asymptotic

text: conﬁdence

text: intervals.

text: Length:

text: the

text: average

text: length

text: of

text: the

text: asymptotic

text: conﬁdence

text: intervals.

text: Size:

text: size

text: of

text: the

text: correct

text: hypothesis

text: H

text: (γ

text: γ

text: γ

text: Power:

text: the

text: probability

text: to

text: reject

text: the

text: false

text: H

text: (γ

text: γ

text: γ

text: estimated.

text: Thus,

text: it

text: is

text: not

text: surprising

text: that

text: the

text: estimation

text: error

text: for

text: α

text: goes

text: down

text: while

text: the

text: one

text: for

text: η

text: increases.

text: The

text: coverage

text: rates

text: of

text: the

text: oracle

text: as

text: well

text: as

text: the

text: desparsiﬁed

text: Lasso

text: improve

text: compared

text: to

text: Table

text: However,

text: the

text: length

text: of

text: the

text: conﬁdence

text: bands

text: does

text: not

text: decrease

text: as

text: much

text: as

text: when

text: T

text: was

text: increased

text: in

text: the

text: previous

text: experiment.

text: The

text: size

text: of

text: the

text: signiﬁcance

text: test

text: decreases

text: when

text: increasing

text: N

text: while

text: power

text: is

text: close

text: to

text: Adding

text: heteroskedasticity

text: has

text: no

text: consequences

text: while

text: the

text: presence

text: of

text: heavy

text: tails

text: has

text: the

text: usual

text: eﬀect.

text: RMSE

text: Coverage

text: Length

text: α

text: η

text: γ

text: γ

text: γ

text: γ

text: γ

text: γ

text: Size

text: Power

text: LS

text: DL

text: Ora

text: LS

text: DL

text: Ora

text: LS

text: DL

text: Ora

text: Table

text: Experiment

text: LS,

text: DL

text: and

text: Ora:

text: least

text: squares

text: including

text: all

text: variables,

text: desparsiﬁed

text: Lasso

text: and

text: least

text: squares

text: oracle.

text: RMSE:

text: root

text: mean

text: square

text: error.

text: Coverage:

text: the

text: coverage

text: rate

text: of

text: the

text: asymptotic

text: conﬁdence

text: intervals.

text: Length:

text: the

text: average

text: length

text: of

text: the

text: asymptotic

text: conﬁdence

text: intervals.

text: Size:

text: size

text: of

text: the

text: correct

text: hypothesis

text: H

text: (γ

text: γ

text: γ

text: Power:

text: the

text: probability

text: to

text: reject

text: the

text: false

text: H

text: (γ

text: γ

text: γ

text: Table

text: contains

text: a

text: setting

text: with

text: more

text: than

text: variables.

text: The

text: main

text: message

text: of

text: the

text: previous

text: tables

text: prevails

text: even

text: in

text: this

text: setting:

text: the

text: coverage

text: of

text: the

text: Lasso-based

text: conﬁdence

text: intervals

text: is

text: almost

text: as

text: good

text: as

text: the

text: ones

text: based

text: on

text: the

text: oracle.

text: On

text: the

text: other

text: hand,

text: the

text: bands

text: of

text: the

text: former

text: are

text: now

text: slightly

text: wider

text: than

text: the

text: ones

text: of

text: the

text: latter.

text: Both

text: procedures

text: have

text: power

text: close

text: to

text: one

text: while

text: the

text: Lasso-based

text: test

text: is

text: a

text: bit

text: oversized

text: compared

text: to

text: the

text: oracle

text: based

text: test.

text: Heteroskedasticity

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text: 7.3 Proof of Lemma 1 (cid:112) The following lemma gives the rates of the uniform prediction and estimation errors for nodewise regression. Lemma 8. Let Assumptions 1, 3 and 4 hold. Let λ = 16M(log p)3/N for some M > 0. node For M suﬃciently large, we have (cid:16) (cid:17) 1 (cid:107) − (cid:107) (cid:16) − (cid:17) max Z− (φˆ φ ) 2 = O G¯λ2 ϑ (7.16) j∈H1 NT j j j p node (cid:107) − (cid:107) − max φˆ φ = O G¯λ1 ϑ (7.17) ∈ j j 1 p node j H1 1 (cid:107) (cid:48) (cid:107) max Z− ζ ∞ = O (λ ). (7.18) j∈H1 NT j j p node − × − Proof. We say that a (p 1) (p 1) matrix A satisﬁes the compatibility condition CC(r) for ∈ { − } some integer r 1, . . . , p 1 if (cid:48) δ Aδ κ2 (A, r) := min min (cid:107) (cid:107) > 0. R⊆{|1,.|.≤.,p−1} (cid:107) δ∈R(cid:107)p−≤1\(cid:107){0}(cid:107) 1r δR 21 R r δRc 1 3 δR 1 − × ∗ Deﬁne a (p 1) 1 vector φ such that j ∗ {| | ≥ } − φ := φ 1 φ λ k = 1, . . . , p 1 j,k j,k j,k node ∗ ∗ and its active set J as well as its sparsity index s j j ∗ { ∗ (cid:54) − } ≤ ∗ | ∗| ≤ − Jj := k : φj,k = 0,(cid:110)k = 1, . . . , p 1 1 sj :(cid:111)= Jj p 1. Consider the events D(cid:26) =(cid:16) max 1 (cid:107)Z−(cid:48) (cid:17)ζ (cid:107)∞ ≤ λnode , (cid:27) N j∈H1 NT j j 4 ∗ EN,j = κ2 N(cid:26)1(cid:13)(cid:13)T Z−(cid:48) jZ−j, s∗j ≥(cid:13)(cid:13) κ2(ΨZ,−2j,(cid:27)−j, sj) , (cid:13) (cid:13) (cid:13) (cid:13) and F 1 (cid:48) − ≤ = Z Z Ψ λ . N Z node NT ∞ Using the same technique as in Section 6.2.3 of Bu¨hlmann and van de Geer (2011), we arrive at the following oracle inequality, which is almost the same as the one on the top of p111 of ∈ D ∩ E Bu¨hlmann and van de Geer (2011): for each j H , on 1 N N,j ∗ 1 (cid:107) − (cid:107) (cid:107) − (cid:107) ≤ 3 (cid:107) ∗ − (cid:107) 48λ2 s (cid:107) ∗ − (cid:107) Z− (φˆ φ ) 2 + λ φˆ φ Z− (φ φ ) 2 + n(cid:48) ode j ∗ + λ φ φ NT j j j node j j 1 NT j j j κ2( 1 Z− Z− , s ) node j j 1 NT j j j ∗ ≤ 3 (cid:107) ∗ − (cid:107) 96λ2 s (cid:107) ∗ − (cid:107) Z− (φ φ ) 2 + node ∗j + λ φ φ NT j j j κ2(Ψ , s ) node j j 1 Z j (cid:0) (cid:1) (7.19) E ≥ where the second inequality is due to event and that κ2 Ψ − − , r κ2 (Ψ , r) for − N,j Z, j, j Z all j = 1, ..., p and r = 1, ..., p 1. 30 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Proof

text: of

text: Lemma

text: The

text: following

text: lemma

text: gives

text: the

text: rates

text: of

text: the

text: uniform

text: prediction

text: and

text: estimation

text: errors

text: for

text: nodewise

text: regression.

text: Lemma

text: Let

text: Assumptions

text: and

text: hold.

text: Let

text: λ

text: for

text: some

text: M

text: node

text: For

text: M

text: suﬃciently

text: large,

text: we

text: have

text: max

text: Z−

text: (φˆ

text: φ

text: O

text: G¯λ2

text: ϑ

text: NT

text: j

text: j

text: j

text: p

text: node

text: max

text: φˆ

text: φ

text: O

text: G¯λ1

text: ϑ

text: j

text: j

text: p

text: node

text: j

text: max

text: Z−

text: ζ

text: O

text: (λ

text: NT

text: j

text: j

text: p

text: node

text: Proof.

text: We

text: say

text: that

text: a

text: (p

text: (p

text: matrix

text: A

text: satisﬁes

text: the

text: compatibility

text: condition

text: CC(r)

text: for

text: some

text: integer

text: r

text: p

text: if

text: δ

text: Aδ

text: κ2

text: (A,

text: r)

text: min

text: min

text: δR

text: R

text: r

text: δRc

text: δR

text: Deﬁne

text: a

text: (p

text: vector

text: φ

text: such

text: that

text: j

text: φ

text: φ

text: φ

text: λ

text: k

text: p

text: j,k

text: j,k

text: j,k

text: node

text: and

text: its

text: active

text: set

text: J

text: as

text: well

text: as

text: its

text: sparsity

text: index

text: s

text: j

text: j

text: Jj

text: k

text: φj,k

text: p

text: sj

text: Jj

text: p

text: Consider

text: the

text: events

text: max

text: λnode

text: N

text: NT

text: j

text: j

text: EN,j

text: κ2

text: jZ−j,

text: s∗j

text: sj)

text: and

text: F

text: Z

text: Z

text: Ψ

text: λ

text: N

text: Z

text: node

text: NT

text: Using

text: the

text: same

text: technique

text: as

text: in

text: Section

text: of

text: Bu¨hlmann

text: and

text: van

text: de

text: Geer

text: we

text: arrive

text: at

text: the

text: following

text: oracle

text: inequality,

text: which

text: is

text: almost

text: the

text: same

text: as

text: the

text: one

text: on

text: the

text: top

text: of

text: of

text: D

text: E

text: Bu¨hlmann

text: and

text: van

text: de

text: Geer

text: for

text: each

text: j

text: H

text: on

text: N

text: N,j

text: s

text: Z−

text: (φˆ

text: φ

text: λ

text: φˆ

text: φ

text: Z−

text: (φ

text: φ

text: ode

text: j

text: λ

text: φ

text: φ

text: NT

text: j

text: j

text: j

text: node

text: j

text: j

text: NT

text: j

text: j

text: j

text: Z−

text: Z−

text: s

text: node

text: j

text: j

text: NT

text: j

text: j

text: j

text: s

text: Z−

text: (φ

text: φ

text: node

text: ∗j

text: λ

text: φ

text: φ

text: NT

text: j

text: j

text: j

text: κ2(Ψ

text: s

text: node

text: j

text: j

text: Z

text: j

text: E

text: where

text: the

text: second

text: inequality

text: is

text: due

text: to

text: event

text: and

text: that

text: κ2

text: Ψ

text: r

text: κ2

text: (Ψ

text: r)

text: for

text: N,j

text: Z,

text: j,

text: j

text: Z

text: all

text: j

text: p

text: and

text: r

text: p

text: ##LTLine##

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text: ##LTLine##

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text: for some 0 < ν < 1 and E = E > 0. Weak sparsity does not require any of the ﬁxed eﬀects to N be zero but instead restricts the ”sum”, E, of all the ﬁxed eﬀects. E can be large in the sense that it tends to inﬁnity but the smaller it is, the sharper will our results be. It is appropriate to stress that the ﬁxed eﬀects can not be entirely unrestricted – that is why our setting is middle ground between random and ﬁxed eﬀects. Thus, our framework also excludes many models of interest. We believe, however, that our results provide a useful ﬁrst step towards uniform inference in high-dimensional dynamic panel data models and we certainly allow for more correlation between η and the covariates than the random eﬀects assumption does. Note that he presence of many control variables in a high-dimensional model leaves less variation to be explained by the unobserved heterogeneities and these are therefore likely to be small in magnitude making the weak sparsity assumption reasonable. Thus, weak sparsity (cid:80) actually becomes more reasonable the larger the number of control variables is. Weak sparsity is a strict generalization of exact sparsity in the sense that if only s elements | | ≤ 2 of η are non-zero and none of these exceeds a constant K, then N η ν s Kν. Thus, i i=1 i 2 E = s Kν works. Alternatively, exact sparsity of η can be handled as the boundary case ν = 0 2 upon deﬁning 00 = 0 such that E will equal the number of non-zero entries of η. 2.3 The Objective Function and Assumptions (cid:48) (cid:48) (cid:48) Our starting point for inference is the minimiser γˆ = (αˆ , ηˆ ) of the following panel Lasso objective function (cid:107) − (cid:107) (cid:107) (cid:107) √λ (cid:107) (cid:107) L(γ) = y Πγ 2 + 2λ α + 2 N η . (2.2) N 1 1 N As usual λ is a positive regularization sequence. Note that we penalize α and η diﬀerently N to reﬂect the fact that we have NT observations to estimate α for j = 1, ..., p while only j T observations are available to estimate each η . Penalizing the ﬁxed eﬀects is not new and i was already done in Koenker (2004) and Galvao and Montes-Rojas (2010) in a low dimensional panel-quantile model. Furthermore, the penalization ﬁts well with the weak sparsity assumption on the ﬁxed eﬀects and may increase eﬃciency of αˆ as found in Galvao and Montes-Rojas (2010). For practical implementation it is very convenient that we only have one penalty parameter λ instead of having separate penalty parameters for α and η. The minimization problem can N be solved easily as it simply corresponds to a weighted Lasso with known weights. However, the probabilistic analysis of the properly scaled Gram matrix is diﬀerent from the one for the standard Lasso as it must be broken into several steps. We now turn to the assumptions needed for our inferential procedure. Assumption 1. { (cid:48) (cid:48) (cid:48) } (x , . . . , x , ε ) N is an independent sequence and i,1 i,T i i=1 E | [ε y − , ..., y − , x , ..., x ] = 0 for i = 1, ..., N, t = 1, ..., T. i,t i,t 1 i,1 L i,t i,1 Assumption 1 imposes independence across i = 1, ..., N which is standard in the panel data literature, see e.g. Wooldridge (2010) or Arellano (2003). Note however, that we do not assume the data to be identically distributed across i = 1, ..., N. Assumption 1 also implies, by iterated expectations, that the error terms form a martingale diﬀerence sequence with respect to the ﬁltration generated by the variables in the above conditioning set and thus restricts the degree of dependence in the error terms across t (in particular, they are uncorrelated).2 However, it still allows for considerable dependence over time, as higher moments than the ﬁrst are not { } 2It can also be veriﬁed that ε T forms a martingale diﬀerence sequence with respect to the natural i,t t=1 ﬁltration for all i = 1, . . . , N. This is because the ε are (linear) functions of the variables in the conditioning i,t set in Assumption 1. 5 ##LTLine## ##LTLine## ##LTLine##

text: for

text: some

text: ν

text: and

text: E

text: E

text: Weak

text: sparsity

text: does

text: not

text: require

text: any

text: of

text: the

text: ﬁxed

text: eﬀects

text: to

text: N

text: be

text: zero

text: but

text: instead

text: restricts

text: the

text: ”sum”,

text: E,

text: of

text: all

text: the

text: ﬁxed

text: eﬀects.

text: E

text: can

text: be

text: large

text: in

text: the

text: sense

text: that

text: it

text: tends

text: to

text: inﬁnity

text: but

text: the

text: smaller

text: it

text: is,

text: the

text: sharper

text: will

text: our

text: results

text: be.

text: It

text: is

text: appropriate

text: to

text: stress

text: that

text: the

text: ﬁxed

text: eﬀects

text: can

text: not

text: be

text: entirely

text: unrestricted

text: that

text: is

text: why

text: our

text: setting

text: is

text: middle

text: ground

text: between

text: random

text: and

text: ﬁxed

text: eﬀects.

text: Thus,

text: our

text: framework

text: also

text: excludes

text: many

text: models

text: of

text: interest.

text: We

text: believe,

text: however,

text: that

text: our

text: results

text: provide

text: a

text: useful

text: ﬁrst

text: step

text: towards

text: uniform

text: inference

text: in

text: high-dimensional

text: dynamic

text: panel

text: data

text: models

text: and

text: we

text: certainly

text: allow

text: for

text: more

text: correlation

text: between

text: η

text: and

text: the

text: covariates

text: than

text: the

text: random

text: eﬀects

text: assumption

text: does.

text: Note

text: that

text: he

text: presence

text: of

text: many

text: control

text: variables

text: in

text: a

text: high-dimensional

text: model

text: leaves

text: less

text: variation

text: to

text: be

text: explained

text: by

text: the

text: unobserved

text: heterogeneities

text: and

text: these

text: are

text: therefore

text: likely

text: to

text: be

text: small

text: in

text: magnitude

text: making

text: the

text: weak

text: sparsity

text: assumption

text: reasonable.

text: Thus,

text: weak

text: sparsity

text: actually

text: becomes

text: more

text: reasonable

text: the

text: larger

text: the

text: number

text: of

text: control

text: variables

text: is.

text: Weak

text: sparsity

text: is

text: a

text: strict

text: generalization

text: of

text: exact

text: sparsity

text: in

text: the

text: sense

text: that

text: if

text: only

text: s

text: elements

text: of

text: η

text: are

text: non-zero

text: and

text: of

text: these

text: exceeds

text: a

text: constant

text: K,

text: then

text: N

text: η

text: ν

text: s

text: Kν.

text: Thus,

text: i

text: i

text: E

text: s

text: Kν

text: works.

text: Alternatively,

text: exact

text: sparsity

text: of

text: η

text: can

text: be

text: handled

text: as

text: the

text: boundary

text: case

text: ν

text: upon

text: deﬁning

text: such

text: that

text: E

text: will

text: equal

text: the

text: number

text: of

text: non-zero

text: entries

text: of

text: η.

text: The

text: Objective

text: Function

text: and

text: Assumptions

text: Our

text: starting

text: point

text: for

text: inference

text: is

text: the

text: minimiser

text: γˆ

text: (αˆ

text: ηˆ

text: of

text: the

text: following

text: panel

text: Lasso

text: objective

text: function

text: √λ

text: L(γ)

text: y

text: Πγ

text: 2λ

text: α

text: N

text: η

text: N

text: N

text: As

text: usual

text: λ

text: is

text: a

text: positive

text: regularization

text: sequence.

text: Note

text: that

text: we

text: penalize

text: α

text: and

text: η

text: diﬀerently

text: N

text: to

text: reﬂect

text: the

text: fact

text: that

text: we

text: have

text: NT

text: observations

text: to

text: estimate

text: α

text: for

text: j

text: p

text: while

text: only

text: j

text: T

text: observations

text: are

text: available

text: to

text: estimate

text: each

text: η

text: Penalizing

text: the

text: ﬁxed

text: eﬀects

text: is

text: not

text: new

text: and

text: i

text: was

text: already

text: done

text: in

text: Koenker

text: and

text: Galvao

text: and

text: Montes-Rojas

text: in

text: a

text: low

text: dimensional

text: panel-quantile

text: model.

text: Furthermore,

text: the

text: penalization

text: ﬁts

text: well

text: with

text: the

text: weak

text: sparsity

text: assumption

text: on

text: the

text: ﬁxed

text: eﬀects

text: and

text: may

text: increase

text: eﬃciency

text: of

text: αˆ

text: as

text: found

text: in

text: Galvao

text: and

text: Montes-Rojas

text: For

text: practical

text: implementation

text: it

text: is

text: very

text: convenient

text: that

text: we

text: only

text: have

text: one

text: penalty

text: parameter

text: λ

text: instead

text: of

text: having

text: separate

text: penalty

text: parameters

text: for

text: α

text: and

text: η.

text: The

text: minimization

text: problem

text: can

text: N

text: be

text: solved

text: easily

text: as

text: it

text: simply

text: corresponds

text: to

text: a

text: weighted

text: Lasso

text: with

text: known

text: weights.

text: However,

text: the

text: probabilistic

text: analysis

text: of

text: the

text: properly

text: scaled

text: Gram

text: matrix

text: is

text: diﬀerent

text: from

text: the

text: one

text: for

text: the

text: standard

text: Lasso

text: as

text: it

text: must

text: be

text: broken

text: into

text: several

text: steps.

text: We

text: now

text: turn

text: to

text: the

text: assumptions

text: needed

text: for

text: our

text: inferential

text: procedure.

text: Assumption

text: (x

text: x

text: ε

text: N

text: is

text: an

text: independent

text: sequence

text: and

text: i,T

text: i

text: E

text: [ε

text: y

text: y

text: x

text: x

text: for

text: i

text: N,

text: t

text: T.

text: i,t

text: i,t

text: L

text: i,t

text: Assumption

text: imposes

text: independence

text: across

text: i

text: N

text: which

text: is

text: standard

text: in

text: the

text: panel

text: data

text: literature,

text: see

text: e.g.

text: Wooldridge

text: or

text: Arellano

text: Note

text: however,

text: that

text: we

text: do

text: not

text: assume

text: the

text: data

text: to

text: be

text: identically

text: distributed

text: across

text: i

text: N.

text: Assumption

text: also

text: implies,

text: by

text: iterated

text: expectations,

text: that

text: the

text: error

text: terms

text: form

text: a

text: martingale

text: diﬀerence

text: sequence

text: with

text: respect

text: to

text: the

text: ﬁltration

text: generated

text: by

text: the

text: variables

text: in

text: the

text: above

text: conditioning

text: set

text: and

text: thus

text: restricts

text: the

text: degree

text: of

text: dependence

text: in

text: the

text: error

text: terms

text: across

text: t

text: (in

text: particular,

text: they

text: are

text: However,

text: it

text: still

text: allows

text: for

text: considerable

text: dependence

text: over

text: time,

text: as

text: higher

text: moments

text: than

text: the

text: ﬁrst

text: are

text: not

text: can

text: also

text: be

text: veriﬁed

text: that

text: ε

text: T

text: forms

text: a

text: martingale

text: diﬀerence

text: sequence

text: with

text: respect

text: to

text: the

text: natural

text: i,t

text: ﬁltration

text: for

text: all

text: i

text: N.

text: This

text: is

text: because

text: the

text: ε

text: are

text: (linear)

text: functions

text: of

text: the

text: variables

text: in

text: the

text: conditioning

text: i,t

text: set

text: in

text: Assumption

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Note that (cid:112) (cid:16) (cid:13) (cid:13) (cid:13) (cid:13) (cid:17) (cid:112) (cid:112) |ρ(cid:48)Θˆ S−1Π(cid:48)ε − ρ(cid:48)ΘS−1Π(cid:48)ε| ≤(cid:13)(cid:107)ρ(cid:48)((cid:13)Θˆ − Θ)(cid:107) (cid:107)(cid:13)S−1Π(cid:13) (cid:48)ε(cid:107)∞ = (cid:107)ρ(cid:48) (Θˆ − Θ )(cid:107) (cid:107)S−1Π(cid:48)ε(cid:107)∞ 1 1 Z Z 1 − √ 1 (cid:48) ∨ √1 (cid:48) − ∨ = O (G¯λ1 ϑ h ) Z ε ∞ D ε ∞ = O (G¯λ1 ϑ h )O ( (log(p N))3) = o (1), p node 1 p node 1 p p NT T where the second equality is due to (7.43), and the third equality is due to (7.12) and (7.13), and the last equality is due to Assumption 6(a)(iii). (cid:48) −→ d 7.4.3 t N(0, 1) 1 (cid:48) We now prove that t is asymptotically distributed as a standard normal by verifying (i)-(iii) 1 (cid:16) (cid:17) (cid:16) (cid:17) (cid:80) (cid:80) (cid:80) of Theorem 5 in Appendix B. Note that (cid:48) − z ε (cid:48) − z ε (cid:48) − (cid:48) ρ ΘS 1 N T i,t i,t ρ ΘS 1 k j j (cid:48) √ρ ΘS 1Π ε √i=1 t=1 d ε √ j=1 d ε i,t i,t j j t := (cid:48) (cid:48) = (cid:48) (cid:48) = (cid:48) (cid:48) , 1 ρ ΘΣ Θ ρ ρ ΘΣ Θ ρ ρ ΘΣ Θ ρ Πε Πε Πε (cid:48) where k := NT . In the proof of Lemma 5, we have shown that t is a martingale diﬀerence 1 array with variance (cid:48) − E (cid:48) (cid:48) − (cid:48) (cid:48) E (cid:48) ρ ΘS 1 [Π εε Π]S 1Θ ρ var(t ) = [t 2] = (cid:48) (cid:48) = 1 1 1 ρ ΘΣ Θ ρ Πε where we have used the deﬁnition of Σ . We have already shown in (7.28) that the denominator (cid:48) Πε (cid:48) of t is uniformly bounded away from zero. Thus, verifying that t satisﬁes (i) and (ii) of 1 1 (cid:48) Theorem 5 in Appendix B is equivalent to verifying that the numerator of t satisﬁes (i) and 1 (ii) of Theorem 5. First, n(cid:13)ote that (cid:13) (cid:13) (cid:13) (cid:13) (cid:88) (cid:13) (cid:88) (cid:13) (cid:13) (cid:113) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:48) (cid:48) ≤ | | (cid:48) − ρ Θ = ρ Θ ρ Θ = O( h G¯λ ϑ ), (7.59) 1 Z 1 1j Z,j 1j Z,j 1 node ∈ ∈ 1 j H1 1 j H1 (cid:12) (cid:16) (cid:17)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:13) (cid:13) (cid:12) (cid:12) w(cid:12) here the last equal(cid:12)ity (cid:12)is due to (7.(cid:12)27)(cid:12). Next,(cid:12) (cid:13) (cid:13) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:107) (cid:107) | | (cid:48) (cid:113) − z ε (cid:12)≤ (cid:48) (cid:12)z√ ε ρ√ε ≤ (cid:48) z√ ε ρ √∞ ε ρ ΘS 1 i,t i,t (cid:12) ρ Θ (cid:12) i,t i,t + 2,i i,t ρ Θ max i,t,l i,t + 2 i,t di,tεi,t (cid:12) 1 Z(cid:12) NT T 1 Z 1 1≤l≤p NT T (cid:107) (cid:107) | | (cid:46) − z√ ε ρ √∞ ε h G¯λ ϑ max i,t,l i,t + 2 i,t , 1 node ≤ ≤ 1 l p NT T where the last inequality due to (7.59). We have already shown in the proof of Lemma 5 that (cid:115) z(cid:13)(cid:113) ε has uniform(cid:12) ly boun(cid:12)ded ψ -Orlicz(cid:13)norm. The same is(cid:13) the case for (cid:13)ε . Hence, (cid:13)i,t,l i,t (cid:12) (cid:12) 1 (cid:13) (cid:13) (cid:13)i,t (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:12) (cid:12) (cid:13) (cid:13) (cid:13) (cid:107) (cid:107) | | − (cid:107) (cid:107) (cid:115) − z√ ε ρ √∞ ε ≤ h G¯λ ϑ (cid:115) ρ√ ∞ h G¯λ ϑ max i,t,l i,t + 2 i,t 1 node max z ε + 2 ε 1 node 1≤l≤p NT (cid:13)(cid:13) T (cid:13)(cid:13) ψ1 (cid:13)(cid:13)NT (cid:13)(cid:13) 1≤l≤p i,t,l i,t ψ1 T i,t ψ1 − (cid:107) (cid:107) − (cid:107) (cid:107) (cid:46) h G¯λ ϑ ρ√ ∞ (cid:46) h G¯λ ϑ ρ√ ∞ 1 node log(1 + p) max z ε + 2 ε 1 node log(1 + p) + 2 , NT 1≤l≤p i,t,l i,t ψ1 T i,t ψ1 NT T for all i and T , where the ﬁrst rate inequality is due to Lemma 2.2.2 in van der Vaart and (cid:115) (cid:34) (cid:35) W(cid:13) ellner (1996(cid:12)). Using(cid:16)Lemma 2(cid:17).(cid:12)2(cid:13).2 in van der Vaart and Wellner (1996) one more time, (cid:13) (cid:12) (cid:12)(cid:13) (cid:13) (cid:12) (cid:12)(cid:13) − (cid:107) (cid:107) (cid:48) − z ε (cid:46) h G¯λ ϑ ρ√ ∞ max max ρ ΘS 1 i,t i,t log(1 + NT ) 1 node log(1 + p) + 2 = o(1), 1≤i≤N 1≤t≤T di,tεi,t ψ1 NT T 46 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Note

text: that

text: (Θˆ

text: Θ

text: Z

text: Z

text: √1

text: O

text: ϑ

text: h

text: Z

text: ε

text: D

text: ε

text: O

text: ϑ

text: h

text: )O

text: (log(p

text: o

text: p

text: node

text: p

text: node

text: p

text: p

text: NT

text: T

text: where

text: the

text: second

text: equality

text: is

text: due

text: to

text: and

text: the

text: third

text: equality

text: is

text: due

text: to

text: and

text: and

text: the

text: last

text: equality

text: is

text: due

text: to

text: Assumption

text: d

text: t

text: We

text: now

text: prove

text: that

text: t

text: is

text: asymptotically

text: distributed

text: as

text: a

text: standard

text: normal

text: by

text: verifying

text: (i)-(iii)

text: of

text: Theorem

text: in

text: Appendix

text: B.

text: Note

text: that

text: z

text: ε

text: z

text: ε

text: ρ

text: ΘS

text: N

text: T

text: i,t

text: i,t

text: ρ

text: ΘS

text: k

text: j

text: j

text: √ρ

text: ΘS

text: 1Π

text: ε

text: d

text: ε

text: d

text: ε

text: i,t

text: i,t

text: j

text: j

text: t

text: ρ

text: ΘΣ

text: Θ

text: ρ

text: ρ

text: ΘΣ

text: Θ

text: ρ

text: ρ

text: ΘΣ

text: Θ

text: ρ

text: Πε

text: Πε

text: Πε

text: where

text: k

text: NT

text: In

text: the

text: proof

text: of

text: Lemma

text: we

text: have

text: shown

text: that

text: t

text: is

text: a

text: martingale

text: diﬀerence

text: array

text: with

text: variance

text: E

text: E

text: ρ

text: ΘS

text: [Π

text: εε

text: Π]S

text: 1Θ

text: ρ

text: var(t

text: [t

text: ρ

text: ΘΣ

text: Θ

text: ρ

text: Πε

text: where

text: we

text: have

text: used

text: the

text: deﬁnition

text: of

text: Σ

text: We

text: have

text: already

text: shown

text: in

text: that

text: the

text: denominator

text: Πε

text: of

text: t

text: is

text: uniformly

text: bounded

text: away

text: from

text: zero.

text: Thus,

text: verifying

text: that

text: t

text: satisﬁes

text: (i)

text: and

text: (ii)

text: of

text: Theorem

text: in

text: Appendix

text: B

text: is

text: equivalent

text: to

text: verifying

text: that

text: the

text: numerator

text: of

text: t

text: satisﬁes

text: (i)

text: and

text: (ii)

text: of

text: Theorem

text: First,

text: that

text: ρ

text: Θ

text: ρ

text: Θ

text: ρ

text: Θ

text: O(

text: h

text: G¯λ

text: ϑ

text: Z

text: Z,j

text: Z,j

text: node

text: j

text: j

text: here

text: the

text: last

text: due

text: to

text: z

text: ε

text: ε

text: ρ√ε

text: z√

text: ε

text: ρ

text: ε

text: ρ

text: ΘS

text: i,t

text: i,t

text: ρ

text: Θ

text: i,t

text: i,t

text: i,t

text: ρ

text: Θ

text: max

text: i,t,l

text: i,t

text: i,t

text: di,tεi,t

text: NT

text: T

text: Z

text: NT

text: T

text: z√

text: ε

text: ρ

text: ε

text: h

text: G¯λ

text: ϑ

text: max

text: i,t,l

text: i,t

text: i,t

text: node

text: l

text: p

text: NT

text: T

text: where

text: the

text: last

text: inequality

text: due

text: to

text: We

text: have

text: already

text: shown

text: in

text: the

text: proof

text: of

text: Lemma

text: that

text: ε

text: has

text: ly

text: ψ

text: The

text: same

text: the

text: case

text: for

text: Hence,

text: i,t

text: z√

text: ε

text: ρ

text: ε

text: h

text: G¯λ

text: ϑ

text: ρ√

text: h

text: G¯λ

text: ϑ

text: max

text: i,t,l

text: i,t

text: i,t

text: node

text: max

text: z

text: ε

text: ε

text: node

text: NT

text: T

text: ψ1

text: i,t,l

text: i,t

text: ψ1

text: T

text: i,t

text: ψ1

text: h

text: G¯λ

text: ϑ

text: ρ√

text: h

text: G¯λ

text: ϑ

text: ρ√

text: node

text: p)

text: max

text: z

text: ε

text: ε

text: node

text: p)

text: NT

text: i,t,l

text: i,t

text: ψ1

text: T

text: i,t

text: ψ1

text: NT

text: T

text: for

text: all

text: i

text: and

text: T

text: where

text: the

text: ﬁrst

text: rate

text: inequality

text: is

text: due

text: to

text: Lemma

text: in

text: van

text: der

text: Vaart

text: and

text: ellner

text: in

text: van

text: der

text: Vaart

text: and

text: Wellner

text: one

text: more

text: time,

text: z

text: ε

text: h

text: G¯λ

text: ϑ

text: ρ√

text: max

text: max

text: ρ

text: ΘS

text: i,t

text: i,t

text: NT

text: node

text: p)

text: di,tεi,t

text: ψ1

text: NT

text: T

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: (cid:32) (cid:48) (cid:33) By the asymptotic normality of t , for N suﬃciently large, (cid:0) (cid:1) (cid:113) (cid:113)1 (cid:48) − (cid:48) (cid:48) P ρ Θˆ S 1Π ε − ρ ∆ ≤ ≥ − − − inf t, A , A , A Φ t(1 (cid:15)) 2D(cid:15) 2(cid:15). ∈F (cid:48) (cid:48) (cid:48) (cid:48) 1,N 2,N 3,N γ (s1,ν,E) ρ Θˆ Σˆ Θˆ ρ ρ Θˆ Σˆ Θˆ ρ Πε Πε (cid:55)→ As the above arguments are valid for every (cid:15) > 0, we can use the continuity of q Φ(q) to conclude that f(cid:32)or every δ > 0, one can choose (cid:15) suﬃciently small suc(cid:33)h that (cid:113) (cid:113) (cid:48) − (cid:48) (cid:48) P ρ Θˆ S 1Π ε − ρ ∆ ≤ ≥ − − inf t, A , A , A Φ (t) δ 2(cid:15). (7.66) ∈F (cid:48) (cid:48) (cid:48) (cid:48) 1,N 2,N 3,N γ (s1,ν,E) ρ Θˆ Σˆ Θˆ ρ ρ Θˆ Σˆ Θˆ ρ Πε Πε P ∪ P ∪ Thus, by (7.65), (7.66) and the fact that sup ∈F ( 3 Ac ) = ( 3 Ac ) = o(1), ∈ R γ (s1,ν,E) i=1 i,N i=1 i,N we have proved (4.1) (the uniformity over t follows from the fact that Φ(t) is continuous). (cid:18)(cid:12) (cid:12) (cid:19) To see (4(cid:16).2), no(cid:104)te that (cid:105)(cid:17) (cid:12) (cid:12) (cid:12) (cid:12) √ (cid:12) (cid:12) − P α ∈/ (cid:18)α˜ − z − √σ˜α,j , α˜ + z −(cid:19) √σ˜(cid:18)α,j = P NT (α˜j αj)(cid:19) > z − j √j 1 δ/2 NT j 1 δ/2 N√T σ˜z,j 1 δ/2 − − ≤ 1 − P NT (α˜j αj) ≤ z − + P NT (α˜j αj) ≤ −z − . σ˜ 1 δ/2 σ˜ 1 δ/2 z,j z,j ∈ F Thus, taking the supremum over γ (s , ν, E) and letting N tend to inﬁnity yields (4.2) via (cid:16)(cid:104) 1 (cid:105)(cid:17) (4.1). The proof is the same for (4.3). Next, we turn to (4.4). √ NT su(cid:16)p(cid:113) diam α˜ − z − √(cid:17)σ˜α,j , α˜ + z(cid:18)−(cid:112) √σ˜α,j (cid:19) ∈F j 1 δ/2 NT j 1 δ/2 NT γ (s1,ν,E) ≤ maxeval(Σ ) = 2z − [Θ Σ Θ ] + o (1) 2z − 1,N + o (1) = O (1), 1 δ/2 Z 1,N Z jj p 1 δ/2 mineval(Ψ ) p p Z where the ﬁrst equality is due to (3.21), and the last equality is due to Assumptions 4(a) and (cid:16)(cid:104) (cid:105)(cid:17) (cid:16)(cid:113) (cid:17) 6(d). Similarly, we can prove (4.5): √ − √σ˜ √σ˜ T sup(cid:18)(cid:20) diam η˜ (cid:21)z − η,i ,(cid:19)η˜ + z − η,i = 2z − [Σ ] + o (1) ∈F (cid:88) i 1 δ/2 T i 1 δ/2 T 1 δ/2 3,N ii p γ (s1,ν,E) 1/2 T 1 E = 2z − [ε2 ] + o (1) = O (1), 1 δ/2 T i,t p p t=1 where the third equality follows from the arguments above (7.53). 8 Appendix B − × − Proposition 1. Let A and B be two positive semideﬁnite (p 1) (p 1) matrices and | − | ∈ { − } δ := max ≤ ≤ − A B . For any integer r 1, . . . , p 1 , one has 1 l,k p 1 lk lk ≥ − κ2(B, r) κ2(A, r) δ16r. Proof. The proof is exactly the same as that of Lemma 6. 50 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: By

text: the

text: asymptotic

text: normality

text: of

text: t

text: for

text: N

text: suﬃciently

text: large,

text: P

text: ρ

text: Θˆ

text: S

text: 1Π

text: ε

text: ρ

text: inf

text: t,

text: A

text: A

text: A

text: Φ

text: ∈F

text: γ

text: ρ

text: Θˆ

text: Σˆ

text: Θˆ

text: ρ

text: ρ

text: Θˆ

text: Σˆ

text: Θˆ

text: ρ

text: Πε

text: Πε

text: As

text: the

text: above

text: arguments

text: are

text: valid

text: for

text: every

text: we

text: can

text: use

text: the

text: continuity

text: of

text: q

text: Φ(q)

text: to

text: conclude

text: that

text: every

text: δ

text: one

text: can

text: choose

text: suﬃciently

text: small

text: that

text: P

text: ρ

text: Θˆ

text: S

text: 1Π

text: ε

text: ρ

text: inf

text: t,

text: A

text: A

text: A

text: Φ

text: (t)

text: δ

text: ∈F

text: γ

text: ρ

text: Θˆ

text: Σˆ

text: Θˆ

text: ρ

text: ρ

text: Θˆ

text: Σˆ

text: Θˆ

text: ρ

text: Πε

text: Πε

text: P

text: P

text: Thus,

text: by

text: and

text: the

text: fact

text: that

text: sup

text: ∈F

text: Ac

text: Ac

text: R

text: γ

text: i,N

text: i,N

text: we

text: have

text: proved

text: (the

text: uniformity

text: over

text: t

text: follows

text: from

text: the

text: fact

text: that

text: Φ(t)

text: is

text: continuous).

text: To

text: see

text: that

text: P

text: α

text: z

text: √σ˜α,j

text: α˜

text: z

text: P

text: NT

text: (α˜j

text: z

text: j

text: √j

text: NT

text: j

text: N√T

text: σ˜z,j

text: P

text: NT

text: (α˜j

text: αj)

text: z

text: P

text: NT

text: (α˜j

text: αj)

text: −z

text: σ˜

text: σ˜

text: z,j

text: z,j

text: F

text: Thus,

text: taking

text: the

text: supremum

text: over

text: γ

text: (s

text: ν,

text: E)

text: and

text: letting

text: N

text: tend

text: to

text: inﬁnity

text: yields

text: via

text: The

text: proof

text: is

text: the

text: same

text: for

text: Next,

text: we

text: turn

text: to

text: NT

text: diam

text: α˜

text: z

text: α˜

text: √σ˜α,j

text: ∈F

text: j

text: NT

text: j

text: NT

text: γ

text: maxeval(Σ

text: [Θ

text: Σ

text: Θ

text: o

text: o

text: O

text: Z

text: Z

text: jj

text: p

text: mineval(Ψ

text: p

text: p

text: Z

text: where

text: the

text: ﬁrst

text: equality

text: is

text: due

text: to

text: and

text: the

text: last

text: equality

text: is

text: due

text: to

text: Assumptions

text: and

text: Similarly,

text: we

text: can

text: prove

text: √σ˜

text: √σ˜

text: T

text: diam

text: η˜

text: η,i

text: z

text: η,i

text: [Σ

text: o

text: ∈F

text: i

text: T

text: i

text: T

text: ii

text: p

text: γ

text: T

text: E

text: o

text: O

text: T

text: i,t

text: p

text: p

text: where

text: the

text: third

text: equality

text: follows

text: from

text: the

text: arguments

text: above

text: Appendix

text: B

text: Proposition

text: Let

text: A

text: and

text: B

text: be

text: two

text: positive

text: semideﬁnite

text: (p

text: (p

text: matrices

text: and

text: δ

text: max

text: A

text: B

text: For

text: any

text: integer

text: r

text: p

text: one

text: has

text: l,k

text: p

text: lk

text: lk

text: r)

text: r)

text: Proof.

text: The

text: proof

text: is

text: exactly

text: the

text: same

text: as

text: that

text: of

text: Lemma

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