text: Proof. See Appendix . Lemma 2 (Boundedness of the derivative of the expected periodogram). The derivative of the expected peri- odogram with respect to ω exists and is bounded in absolute value independently of θ and N. Proof. See appendix .   In analogue to Taniguchi (1979) for stationa(cid:88)ry processes, we introduce the following quantity,   (cid:101) D(N) (γ, f) = 1 log S(N)(ω; γ) + (cid:101)f(ω) , N ∈ X S(N)(ω; γ) ω ΩN X ∈ for all positive integer N, γ Θ and non-negative real-valued function f deﬁned on Ω . We also deﬁne N T(N)(f) = arg min D(N) (γ, f) . ∈ γ Θ (cid:55)→ This minimum for ﬁxed f is well deﬁned since the set Θ is compact and since the function γ D(N) (γ, f) (cid:16) (cid:17) is continuous. However in cases where the minimum is reached not uniquely but at multiple parameter values, T(N)(f) will denote any of these values, chosen arbitrarily. Note that, by the deﬁnition of our frequency domain (cid:101) · estimator, we have θˆ(N) = T(N) Sˆ(N)( ) . We start with three lemmas that will be required in proving Theorem X 1 which establishes consistency. (cid:101) Lemma 3. We have, for N large enough, T(N)(S(N)(ω; θ)) = θ, uniquely. X (cid:16) (cid:17) Proof. See Appendix . (cid:101) → · (N) This shows that for all N large enough, the function γ D γ, S ( ; θ) reaches a global minimum at the (cid:101) X · (N) true parameter vector θ. However because S ( ; θ) is changing with N an(cid:16)d is not expec(cid:17)ted to(cid:16)converge to (cid:17)a X given function, we need the following stronger result. (cid:101) (cid:101) { } ∈ N (N) · − (N) · Lemma 4. If γ ∈N Θ is a sequence of parameter vectors such that D γ , S ( ; θ) D θ, S ( ; θ) N N N X X converges to zero when N goes to inﬁnity, then γ converges to θ. N (cid:16) (cid:17) (cid:16) (cid:17) Proof. See Appendix . (cid:101) (cid:101) · · We now show that the functions D γ, S(N)( ; θ) and D γ, Sˆ(N)( ) , deﬁned on Θ, behave asymptotically in X X the same way. Fo(cid:8)r this, we ﬁrst need th(cid:9)e following lemma where we bound the asymptotic variance of some linear functionals of the periodogram. ∈ − (cid:40) (cid:41) (cid:18) (cid:19) Lemma 5. Let a(N)(ω) : ω [ π, π) ∈N(cid:88)be a family of real-valued functions, uniformly bounded by a positive N real number. We have (cid:101) 1 O 1 var a(N)(ω)Sˆ(N)(ω) = . N ∈ X N (cid:110) ω ΩN (cid:111) Proof. See Appendix (cid:101) (cid:101) (cid:18) (cid:19) Remembering that S(N(cid:88))(ω; θ) = E Sˆ(N)(ω); θ(cid:88), we thus have X X a(N)(ω)Sˆ((cid:101)N)(ω) = a(N)(ω)S((cid:101)N)(ω; θ) + O √1 . ∈ X ∈ X P N ω ΩN ω ΩN We are now able to state a consistency theorem for our estimator θˆ(N). (cid:16) (cid:17) (cid:16) (cid:17) −→ Theorem 1 (Consistency of the frequency domain estimator). We have θˆ(N) P θ in probability. (cid:101) (cid:101) Proof. The proof is based on Taniguchi (1979). Denote h(N)(γ; θ) = D γ, S(N)(ω; θ) and hˆ(N)(γ) = D γ, Sˆ(N)(ω) ∈ X X deﬁned for any γ Θ. We have, 22 ##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: Proof.

text: See

text: Appendix

text: Lemma

text: (Boundedness

text: of

text: the

text: derivative

text: of

text: the

text: expected

text: periodogram).

text: The

text: derivative

text: of

text: the

text: expected

text: peri-

text: odogram

text: with

text: respect

text: to

text: ω

text: exists

text: and

text: is

text: bounded

text: in

text: absolute

text: value

text: independently

text: of

text: θ

text: and

text: N.

text: Proof.

text: See

text: appendix

text: In

text: analogue

text: to

text: Taniguchi

text: for

text: processes,

text: we

text: introduce

text: the

text: following

text: quantity,

text: D(N)

text: (γ,

text: f)

text: log

text: S(N)(ω;

text: γ)

text: N

text: X

text: S(N)(ω;

text: γ)

text: ω

text: ΩN

text: X

text: for

text: all

text: positive

text: integer

text: N,

text: γ

text: Θ

text: and

text: non-negative

text: real-valued

text: function

text: f

text: deﬁned

text: on

text: Ω

text: We

text: also

text: deﬁne

text: N

text: T(N)(f)

text: arg

text: min

text: D(N)

text: (γ,

text: f)

text: γ

text: Θ

text: This

text: minimum

text: for

text: ﬁxed

text: f

text: is

text: well

text: deﬁned

text: since

text: the

text: set

text: Θ

text: is

text: compact

text: and

text: since

text: the

text: function

text: γ

text: D(N)

text: (γ,

text: f)

text: is

text: continuous.

text: However

text: in

text: cases

text: where

text: the

text: minimum

text: is

text: reached

text: not

text: uniquely

text: but

text: at

text: multiple

text: parameter

text: values,

text: T(N)(f)

text: will

text: denote

text: any

text: of

text: these

text: values,

text: chosen

text: arbitrarily.

text: Note

text: that,

text: by

text: the

text: deﬁnition

text: of

text: our

text: frequency

text: domain

text: estimator,

text: we

text: have

text: θˆ(N)

text: T(N)

text: Sˆ(N)(

text: We

text: start

text: with

text: three

text: lemmas

text: that

text: will

text: be

text: required

text: in

text: proving

text: Theorem

text: X

text: which

text: establishes

text: consistency.

text: Lemma

text: We

text: have,

text: for

text: N

text: large

text: enough,

text: T(N)(S(N)(ω;

text: θ))

text: θ,

text: uniquely.

text: X

text: Proof.

text: See

text: Appendix

text: (N)

text: This

text: shows

text: that

text: for

text: all

text: N

text: large

text: enough,

text: the

text: function

text: γ

text: D

text: γ,

text: S

text: θ)

text: reaches

text: a

text: global

text: minimum

text: at

text: the

text: X

text: (N)

text: true

text: parameter

text: vector

text: θ.

text: However

text: because

text: S

text: θ)

text: is

text: changing

text: with

text: N

text: is

text: not

text: to

text: X

text: given

text: function,

text: we

text: need

text: the

text: following

text: stronger

text: result.

text: N

text: (N)

text: (N)

text: Lemma

text: If

text: γ

text: ∈N

text: Θ

text: is

text: a

text: sequence

text: of

text: parameter

text: vectors

text: such

text: that

text: D

text: γ

text: S

text: θ)

text: D

text: θ,

text: S

text: θ)

text: N

text: N

text: N

text: X

text: X

text: converges

text: to

text: zero

text: when

text: N

text: goes

text: to

text: inﬁnity,

text: then

text: γ

text: converges

text: to

text: θ.

text: N

text: Proof.

text: See

text: Appendix

text: We

text: now

text: show

text: that

text: the

text: functions

text: D

text: γ,

text: S(N)(

text: θ)

text: and

text: D

text: γ,

text: Sˆ(N)(

text: deﬁned

text: on

text: Θ,

text: behave

text: asymptotically

text: in

text: X

text: X

text: the

text: same

text: way.

text: this,

text: we

text: ﬁrst

text: need

text: following

text: lemma

text: where

text: we

text: bound

text: the

text: asymptotic

text: variance

text: of

text: some

text: linear

text: functionals

text: of

text: the

text: periodogram.

text: Lemma

text: Let

text: a(N)(ω)

text: ω

text: π,

text: π)

text: a

text: family

text: of

text: real-valued

text: functions,

text: uniformly

text: bounded

text: by

text: a

text: positive

text: N

text: real

text: number.

text: We

text: have

text: O

text: var

text: a(N)(ω)Sˆ(N)(ω)

text: N

text: X

text: N

text: ω

text: ΩN

text: Proof.

text: See

text: Appendix

text: Remembering

text: that

text: θ)

text: E

text: Sˆ(N)(ω);

text: we

text: thus

text: have

text: X

text: X

text: θ)

text: O

text: √1

text: X

text: X

text: P

text: N

text: ω

text: ΩN

text: ω

text: ΩN

text: We

text: are

text: now

text: able

text: to

text: state

text: a

text: consistency

text: theorem

text: for

text: our

text: estimator

text: θˆ(N).

text: Theorem

text: (Consistency

text: of

text: the

text: frequency

text: domain

text: estimator).

text: We

text: have

text: θˆ(N)

text: P

text: θ

text: in

text: probability.

text: Proof.

text: The

text: proof

text: is

text: based

text: on

text: Taniguchi

text: Denote

text: h(N)(γ;

text: θ)

text: D

text: γ,

text: S(N)(ω;

text: θ)

text: and

text: hˆ(N)(γ)

text: D

text: γ,

text: Sˆ(N)(ω)

text: X

text: X

text: deﬁned

text: for

text: any

text: γ

text: Θ.

text: We

text: have,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:88) (cid:12) (cid:12) (cid:88) (cid:12) (cid:12) (cid:88) (cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:80) (cid:12)(cid:12) (cid:12)(cid:12) (cid:80) (cid:12)(cid:12) (cid:12) − − (cid:12) (cid:12) − − (cid:12) (cid:12) − − (cid:12) N1 Nt=10 τ gt∗gt+τ = (cid:12)(cid:12)(cid:12)(cid:12)N1 Nt=10 τ(cid:88)ei tu+=τ(cid:80)t−1 βu = N1(cid:12)(cid:12)(cid:12)(cid:12)Nt=10 τ e(cid:12)(cid:12)(cid:12)(cid:12)i (cid:40)tu+=τt(cid:88)−1(Ξ+β(cid:80)u−Ξ) (cid:41)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12) − − (cid:12) (cid:12) − − (cid:12) = N1(cid:12)(cid:12)(cid:12)eiτ(cid:88)Ξ N 1 τ(cid:40)ei (cid:88)tu+=τt−1(βu−Ξ)(cid:41)(cid:12)(cid:12)(cid:12)≥ N1 (cid:60) N 1 τ ei tu+=τt−1(βu−Ξ) (cid:12) t=0 (cid:12) t=0 (cid:12) − − − (cid:12) 1 N 1 τ t+τ 1 − = (cid:12)(cid:12)(cid:80)N cos (cid:12)(cid:12) (cid:80) (βu Ξ) . (cid:12) t=0 (cid:12) u=t − − ≤ − | − | ≤ Using the triangle inequ(cid:12)ality it follows (cid:12)tu+=τt 1 βu Ξ tu+=τt 1 βu Ξ τ∆, the fact that τ∆ < π2 by assumption, and that the cosine function(cid:12)(cid:12)(cid:12) is de(cid:88)creasing on t(cid:12)(cid:12)(cid:12)he interv(cid:88)al [0, π2 ] we obtain (cid:12) − − (cid:12) − − N1 Nt=10 τ gt∗gt+τ ≥ N1 Nt=10 τ cos(τ∆) = (1 − Nτ )cos(τ∆) N→→∞ cos(τ∆) > 0(cid:12)(cid:12)(cid:12), (cid:12)(cid:12)(cid:12) ≤ ≤ as 0 τβ < τ 2πL ∈ π2(cid:101). The above converges to a non-zero value as N goes to inﬁnity, so that lNim→i∞nf c(gN)(τ) > 0. It is true in particular for τ Γ. This shows that the process Zt is a modulated process with a signiﬁcant correlation contribution. A.4 Proof of Lemma 1 Proof. We denote SX,max = maxθ,ω SX(ω;θ) and SX,min = minθ,ω SX(ω;θ). (cid:90) 1. We ﬁrst show the existence of the upper bound. According to Proposition 1 the expected periodogram can be expressed, for ∈ − ∈ ∈ N ω [ π,π], θ Θ and N , by S(X(cid:101)N)(ω;θ) = 2(cid:90)π −ππ SX(ω − λ)Sg(N)(λ)dλ. (cid:88) Therefore, − S(X(cid:101)N)(ω;θ) ≤ 2πSX,max −ππ Sg(N)(λ)dλ = SX,max N1 Nt=01|gt|2, by Parseval equality, and ﬁnally, (cid:101) S(XN)(ω;θ) ≤ gm2 axSX,max, and this by assumption is ﬁnite. 2. Similarly, we show the existence of a lower bound. According to the assumption of a modulated process with signiﬁcant ∈ correlati≥on contribution, there exists a non-negative integer τ Γ and a positive real number ατ such that for N large enough, c(gN)(τ) ατ. Then, (cid:90) (cid:88) − S(X(cid:101)N)(ω;θ) ≥ 2πSX,min −(cid:88)ππ Sg(N)(λ)dλ = SX,min(cid:118)(cid:117)(cid:117)(cid:116)N1 Nt(cid:88)=01|gt|2 (cid:118)(cid:117)(cid:117)(cid:116) (cid:88) − − − − − ≥ 1 N τ 1| | ≥ 1 N τ 1| | N 1| | SX,min N (cid:12)(cid:12)(cid:12) (cid:88) gt 2 (cid:12)(cid:12)(cid:12) SX,min N gt 2 gt 2 (cid:12) t=0 (cid:12) t=0 t=τ (cid:12) − − (cid:12) ≥ 1 N τ 1 ∗ SX,min N gt gt+τ , t=0 (cid:101) (N) ≥ by Cauchy-Schwartz inequality. Hence we get for N large enough SX (ω;θ) ατSX,min. This proves the stated result. A.5 Proof of Lemma 2 ∈ − ∈ ∈ N (cid:90) ((cid:101)N) Proof. We have for all ω [ π,π], θ Θ and N that the form of S (ω;θ) is given by X S(X(cid:101)N)(ω;θ) = 2π −π SX(ω − λ;θ)Sg(N)(λ)dλ. π → We(cid:12)(cid:12)obtain (where(cid:12)(cid:12) the in(cid:12)v(cid:90)ersion of diﬀerentiation and in(cid:12)tegration is a(cid:26)c(cid:12)onsequence(cid:12)o(cid:27)f (cid:90)the diﬀerentiability of the func(cid:26)ti(cid:12)ons ω SX(cid:12)(cid:27)(ω;θ) and(cid:12)the fact that(cid:12)the sp(cid:12)ectral densities are bounded ab(cid:12)ove), (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:101) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (N) (cid:12) ∂S∂Xω (ω;θ) = 2π −ππ ∂∂SωX (ω − λ;θ)Sg(N)(λ)dλ ≤ 2π mλ,aθx ∂∂SωX (λ;θ) −ππ Sg(N)(λ)dλ ≤ gm2 ax mω,aθx ∂∂SωX (ω;θ) , which concludes the proof. 28 ##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##

label: equation

text: τ

text: gt∗gt+τ

text: βu

text: τ

text: N

text: N

text: τ

text: ei

text: N

text: τ

text: t+τ

label: list

text: cos

text: (βu

text: Ξ)

text: u=t

text: Using

text: the

text: triangle

text: it

text: follows

text: βu

text: Ξ

text: tu+=τt

text: βu

text: Ξ

text: τ∆,

text: the

text: fact

text: that

text: τ∆

text: π2

text: by

text: assumption,

text: and

text: that

text: the

text: cosine

text: is

text: on

text: π2

text: we

text: obtain

text: τ

text: gt∗gt+τ

text: τ

text: cos(τ∆)

text: Nτ

text: )cos(τ∆)

text: N→→∞

text: cos(τ∆)

text: as

text: τβ

text: τ

text: The

text: above

text: converges

text: to

text: a

text: non-zero

text: value

text: as

text: N

text: goes

text: to

text: inﬁnity,

text: so

text: that

text: lNim→i∞nf

text: c(gN)(τ)

text: It

text: is

text: true

text: in

text: particular

text: for

text: τ

text: Γ.

text: This

text: shows

text: that

text: the

text: process

text: Zt

text: is

text: a

text: modulated

text: process

text: with

text: a

text: signiﬁcant

text: correlation

text: contribution.

text: Proof

text: of

text: Lemma

text: Proof.

text: We

text: denote

text: SX,max

text: maxθ,ω

text: SX(ω;θ)

text: and

text: SX,min

text: minθ,ω

text: SX(ω;θ).

text: We

text: ﬁrst

text: show

text: the

text: existence

text: of

text: the

text: upper

text: bound.

text: According

text: to

text: Proposition

text: the

text: expected

text: periodogram

text: can

text: be

text: expressed,

text: for

text: N

text: ω

text: π,π],

text: θ

text: Θ

text: and

text: N

text: by

text: −ππ

text: SX(ω

text: λ)Sg(N)(λ)dλ.

text: Therefore,

text: −ππ

text: Sg(N)(λ)dλ

text: SX,max

text: by

text: Parseval

text: equality,

text: and

text: ﬁnally,

text: S(XN)(ω;θ)

text: axSX,max,

text: and

text: this

text: by

text: assumption

text: is

text: ﬁnite.

text: Similarly,

text: we

text: show

text: the

text: existence

text: of

text: a

text: lower

text: bound.

text: According

text: to

text: the

text: assumption

text: of

text: a

text: modulated

text: process

text: with

text: signiﬁcant

text: correlati≥on

text: contribution,

text: there

text: exists

text: a

text: non-negative

text: integer

text: τ

text: Γ

text: and

text: a

text: positive

text: real

text: number

text: ατ

text: such

text: that

text: for

text: N

text: large

text: enough,

text: c(gN)(τ)

text: ατ.

text: Then,

text: Sg(N)(λ)dλ

text: N

text: τ

text: N

text: τ

text: N

text: SX,min

text: N

text: gt

text: SX,min

text: N

text: gt

text: gt

text: t=τ

text: N

text: τ

text: SX,min

text: N

text: gt

text: gt+τ

text: (N)

text: by

text: Cauchy-Schwartz

text: inequality.

text: Hence

text: we

text: get

text: for

text: N

text: large

text: enough

text: SX

text: (ω;θ)

text: ατSX,min.

text: This

text: proves

text: the

text: stated

text: result.

text: Proof

text: of

text: Lemma

text: N

text: Proof.

text: We

text: have

text: for

text: all

text: ω

text: π,π],

text: θ

text: Θ

text: and

text: N

text: that

text: the

text: form

text: of

text: S

text: (ω;θ)

text: is

text: given

text: by

text: X

text: 2π

text: −π

text: SX(ω

text: λ;θ)Sg(N)(λ)dλ.

text: π

text: the

text: of

text: diﬀerentiation

text: and

text: is

text: diﬀerentiability

text: of

text: the

text: ω

text: fact

text: densities

text: are

text: bounded

text: (N)

text: ∂S∂Xω

text: (ω;θ)

text: 2π

text: −ππ

text: ∂∂SωX

text: (ω

text: λ;θ)Sg(N)(λ)dλ

text: 2π

text: mλ,aθx

text: ∂∂SωX

text: (λ;θ)

text: −ππ

text: Sg(N)(λ)dλ

text: ax

text: mω,aθx

text: ∂∂SωX

text: (ω;θ)

text: which

text: concludes

text: the

text: proof.

text: ##LTLine##

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text: The deﬁnition of a signiﬁca(cid:101)nt correlation contribution constrains how much energy adds up for any ﬁxed lag ∈ τ Γ. We see directly from (3) that if we assume a signiﬁcant correlation contribution, the expectation of the estimated autocovariance of X does not vanish with the length of the observation N, at least for lags in Γ. t This allows for consistent estimation of the parameter θ as we will see in Section .(cid:101)As a trivial counterexample, assume for instance that c(N)(τ) goes to zero when N goes to inﬁnity. Then cˆ(N)(τ) in (2) goes to zero as g X well, independently of the parameter vector θ, resulting either in infeasible estimation or requiring a change of estimation approach. Asymptotically stationary modulated processes are a subclass of modulated processes with a signiﬁcant corre- lation contribution. Speciﬁcally, for the class of asymptotically stationary modulated processes, (8) requires that c(N)(τ) converges to the non-zero quantity R (τ), which is a stronger requirement than (9) where we only require g g an asymptotic positive lower bound rather than convergence. 2.2 Missing observations A partic{ularly ∈enNti}cing use of modulated p(cid:26)rocesses is to account fo∈r Nmissing observations in stationary time series. Let X : t be a stationary process. For each time point t , we set (Parzen, 1963), t 0 if X is missing g = t . (12) (cid:101) t 1 if X is observed t ∈ N The process X = g X is formed at all time points t , forming a modulated process in the sense of t t t Deﬁnition 1. An example where the missing observation pattern is deterministic and leads to an asymptotically stationary modulated process is the case of (k, l)-periodically missing data treated by Jones (1962) and Parzen (1963). This corresponds to observing the k ﬁrst values, missing the l next values, observing the k next values, and so on. Note that Parzen (1963) requires k > l for non-parametric estimation of the spectral density of X based on (8). ≤ t Our model of modulated processes with signiﬁcant correlation contribution allows for k l, as long as we observe the lags in Γ. A generalization of this missing data scheme was introduced by Clinger and Ness (1976) with an application to oceanography. Missing observations can also occur according to a random mechanism. This can be modelled by a random modulation sequence taking values zero and one (Scheinok, 1965, Bloomﬁeld, 1970), when the random mechanism according to which missing points occur is independent from the observed process, which we shall assume. Condi- tioning on the observed modulation function, we then return to the deterministic modulating sequence described in this paper. Most works, to our knowledge, have assumed some sort of stationarity for the random modulation sequence, i.e. that the sample autocovariance of the modulation sequence converges almost surely to a non-zero value at all lags (Dunsmuir and Robinson, 1981c,a). Some authors do not require such an assumption but have treated only speciﬁc models, usually autoregressive models (Jones, 1980, Broersen et al., 2004). The deﬁnition of a modulated process with a signiﬁcant correlation contribution in such a situation needs to be understood in a probabilistic fashion, i.e. we require that Property 2 of Deﬁnition 3 be satisﬁed with probability one. Indeed, if one sees th(cid:101)e general random experiment as a two-step experiment, where ﬁrst the random mod(cid:101)ulating sequence { } { } g is generated and observed and then a stationary process X is modulated by this modulating sequ(cid:101)ence to t { } t { } { } produce X , then with probability one the moduating sequence g in the ﬁrst step makes X a modulated t t t { } process with signiﬁcant correlation contribution. Such a situation will be described by saying that X is a t modulated process with an almost surely signiﬁcant correlation contribution. We shall now give a few examples of cases satisfying the stated conditions. { · · · } 1. Let X be an AR(p) Gaussian process with mean zero. If we set Γ = 0, , p , and if the missing data t ≥ occurs deterministically according to a (k, l)-periodic pattern, k p is a suﬃcient condition for the resulting modulated process to have a signiﬁcant correlation contribution. This is because we are able to observe an inﬁnite number of time the lags in Γ. We do not require any additional condition on l. 2. Let X be an AR(p) process, and consider the missing data scheme treated by Scheinok (1965), where t (cid:12) (cid:12) the random mechanism is a sequence of Bernoulli i.i(cid:12).d trials(cid:12) with identical probability of success (to be ≤ (cid:12) (cid:12) ∈ N understood as observation here) 0 < p 1. According to the strong central limit theorem, for all lag τ , c(N)(τ) converges a.s. to p2 > 0 and therefore lim inf c(N)(τ) > 0 a.s. Therefore the observed process is a g →∞ g N modulated process with an almost surely signiﬁcant correlation contribution. { } 3. Consider the random mechanism where the sequence g is generated according to t ∼ B g (p ), (13) t t 6

text: The

label: paragraph

text: deﬁnition

text: of

text: a

text: correlation

text: contribution

text: constrains

text: how

text: much

text: energy

text: adds

text: up

text: for

text: any

text: ﬁxed

text: lag

text: τ

text: Γ.

text: We

text: see

text: directly

text: from

text: that

text: if

text: we

text: assume

text: a

text: signiﬁcant

text: correlation

text: contribution,

text: the

text: expectation

text: of

text: the

text: estimated

text: autocovariance

text: of

text: X

text: does

text: not

text: vanish

text: with

text: the

text: length

text: of

text: the

text: observation

text: N,

text: at

text: least

text: for

text: lags

text: in

text: Γ.

text: t

text: This

text: allows

text: for

text: consistent

text: estimation

text: of

text: the

text: parameter

text: θ

text: as

text: we

text: will

text: see

text: in

text: Section

text: a

text: trivial

text: counterexample,

text: assume

text: for

text: instance

text: that

text: c(N)(τ)

text: goes

text: to

text: zero

text: when

text: N

text: goes

text: to

text: inﬁnity.

text: Then

text: cˆ(N)(τ)

text: in

text: goes

text: to

text: zero

text: as

text: g

text: X

text: well,

text: independently

text: of

text: the

text: parameter

text: vector

text: θ,

text: resulting

text: either

text: in

text: infeasible

text: estimation

text: or

text: requiring

text: a

text: change

text: of

text: estimation

text: approach.

text: Asymptotically

text: stationary

text: modulated

text: processes

text: are

text: a

text: subclass

text: of

text: modulated

text: processes

text: with

text: a

text: signiﬁcant

text: corre-

text: lation

text: contribution.

text: Speciﬁcally,

text: for

text: the

text: class

text: of

text: asymptotically

text: stationary

text: modulated

text: processes,

text: requires

text: that

text: c(N)(τ)

text: converges

text: to

text: the

text: non-zero

text: quantity

text: R

text: (τ),

text: which

text: is

text: a

text: stronger

text: requirement

text: than

text: where

text: we

text: only

text: require

text: g

text: g

text: an

text: asymptotic

text: positive

text: lower

text: bound

text: rather

text: than

text: convergence.

text: Missing

text: observations

text: A

text: partic{ularly

text: ∈enNti}cing

text: use

text: of

text: modulated

text: is

text: to

text: account

text: fo∈r

text: Nmissing

text: observations

text: in

text: stationary

text: time

text: series.

text: Let

text: X

text: t

text: be

text: a

text: stationary

text: process.

text: For

text: each

text: time

text: point

text: t

text: we

text: set

text: (Parzen,

text: t

text: if

text: X

text: is

text: missing

text: g

text: t

text: t

text: if

text: X

text: is

text: observed

text: t

text: N

text: The

text: process

text: X

text: g

text: X

text: is

text: formed

text: at

text: all

text: time

text: points

text: t

text: forming

text: a

text: modulated

text: process

text: in

text: the

text: sense

text: of

text: t

text: t

text: t

text: Deﬁnition

text: An

text: example

text: where

text: the

text: missing

text: observation

text: pattern

text: is

text: deterministic

text: and

text: leads

text: to

text: an

text: asymptotically

text: stationary

text: modulated

text: process

text: is

text: the

text: case

text: of

text: (k,

text: l)-periodically

text: missing

text: data

text: treated

text: by

text: Jones

text: and

text: Parzen

text: This

text: corresponds

text: to

text: observing

text: the

text: k

text: ﬁrst

text: values,

text: missing

text: the

text: l

text: next

text: values,

text: observing

text: the

text: k

text: next

text: values,

text: and

text: so

text: on.

text: Note

text: that

text: Parzen

text: requires

text: k

text: l

text: for

text: non-parametric

text: estimation

text: of

text: the

text: spectral

text: density

text: of

text: X

text: based

text: on

text: t

text: Our

text: model

text: of

text: modulated

text: processes

text: with

text: signiﬁcant

text: correlation

text: contribution

text: allows

text: for

text: k

text: l,

text: as

text: long

text: as

text: we

text: observe

text: the

text: lags

text: in

text: Γ.

text: A

text: generalization

text: of

text: this

text: missing

text: data

text: scheme

text: was

text: introduced

text: by

text: Clinger

text: and

text: Ness

text: with

text: an

text: application

text: to

text: oceanography.

text: Missing

text: observations

text: can

text: also

text: occur

text: according

text: to

text: a

text: random

text: mechanism.

text: This

text: can

text: be

text: modelled

text: by

text: a

text: random

text: modulation

text: sequence

text: taking

text: values

text: zero

text: and

text: one

text: (Scheinok,

text: Bloomﬁeld,

text: when

text: the

text: random

text: mechanism

text: according

text: to

text: which

text: missing

text: points

text: occur

text: is

text: independent

text: from

text: the

text: observed

text: process,

text: which

text: we

text: shall

text: assume.

text: Condi-

text: tioning

text: on

text: the

text: observed

text: modulation

text: function,

text: we

text: then

text: return

text: to

text: the

text: deterministic

text: modulating

text: sequence

text: described

text: in

text: this

text: paper.

text: Most

text: works,

text: to

text: our

text: knowledge,

text: have

text: assumed

text: some

text: sort

text: of

text: stationarity

text: for

text: the

text: random

text: modulation

text: sequence,

text: i.e.

text: that

text: the

text: sample

text: autocovariance

text: of

text: the

text: modulation

text: sequence

text: converges

text: almost

text: surely

text: to

text: a

text: non-zero

text: value

text: at

text: all

text: lags

text: (Dunsmuir

text: and

text: Robinson,

text: Some

text: authors

text: do

text: not

text: require

text: such

text: an

text: assumption

text: but

text: have

text: treated

text: only

text: speciﬁc

text: models,

text: usually

text: autoregressive

text: models

text: (Jones,

text: Broersen

text: et

text: al.,

text: The

text: deﬁnition

text: of

text: a

text: modulated

text: process

text: with

text: a

text: signiﬁcant

text: correlation

text: contribution

text: in

text: such

text: a

text: situation

text: needs

text: to

text: be

text: understood

text: in

text: a

text: probabilistic

text: fashion,

text: i.e.

text: we

text: require

text: that

text: Property

text: of

text: Deﬁnition

text: be

text: satisﬁed

text: with

text: probability

text: one.

text: Indeed,

text: if

text: one

text: sees

text: general

text: random

text: experiment

text: as

text: a

text: two-step

text: experiment,

text: where

text: ﬁrst

text: the

text: random

text: sequence

text: g

text: is

text: generated

text: and

text: observed

text: and

text: then

text: a

text: stationary

text: process

text: X

text: is

text: modulated

text: by

text: this

text: modulating

text: to

text: t

text: t

text: produce

text: X

text: then

text: with

text: probability

text: one

text: the

text: moduating

text: sequence

text: g

text: in

text: the

text: ﬁrst

text: step

text: makes

text: X

text: a

text: modulated

text: t

text: t

text: t

text: process

text: with

text: signiﬁcant

text: correlation

text: contribution.

text: Such

text: a

text: situation

text: will

text: be

text: described

text: by

text: saying

text: that

text: X

text: is

text: a

text: t

text: modulated

text: process

text: with

text: an

text: almost

text: surely

text: signiﬁcant

text: correlation

text: contribution.

text: We

text: shall

text: now

text: give

text: a

text: few

text: examples

text: of

text: cases

text: satisfying

text: the

text: stated

text: conditions.

text: Let

text: X

text: be

text: an

text: AR(p)

text: Gaussian

text: process

text: with

text: mean

text: zero.

text: If

text: we

text: set

text: Γ

text: p

text: and

text: if

text: the

text: missing

text: data

text: t

text: occurs

text: deterministically

text: according

text: to

text: a

text: (k,

text: l)-periodic

text: pattern,

text: k

text: p

text: is

text: a

text: suﬃcient

text: condition

text: for

text: the

text: resulting

text: modulated

text: process

text: to

text: have

text: a

text: signiﬁcant

text: correlation

text: contribution.

text: This

text: is

text: because

text: we

text: are

text: able

text: to

text: observe

text: an

text: inﬁnite

text: number

text: of

text: time

text: the

text: lags

text: in

text: Γ.

text: We

text: do

text: not

text: require

text: any

text: additional

text: condition

text: on

text: l.

text: Let

text: X

text: be

text: an

text: AR(p)

text: process,

text: and

text: consider

text: the

text: missing

text: data

text: scheme

text: treated

text: by

text: Scheinok

text: where

text: t

text: the

text: random

text: mechanism

text: is

text: a

text: sequence

text: of

text: Bernoulli

text: with

text: identical

text: probability

text: of

text: success

text: (to

text: be

text: N

text: understood

text: as

text: observation

text: here)

text: p

text: According

text: to

text: the

text: strong

text: central

text: limit

text: theorem,

text: for

text: all

text: lag

text: τ

text: c(N)(τ)

text: converges

text: a.s.

text: to

text: and

text: therefore

text: lim

text: inf

text: c(N)(τ)

text: a.s.

text: Therefore

text: the

text: observed

text: process

text: is

text: a

text: g

text: g

text: N

text: modulated

text: process

text: with

text: an

text: almost

text: surely

text: signiﬁcant

text: correlation

text: contribution.

text: Consider

text: the

text: random

text: mechanism

text: where

text: the

text: sequence

text: g

text: is

text: generated

text: according

text: to

text: t

text: B

text: g

text: (p

text: t

text: t


text: (cid:101) (cid:101) ∈ Proof. Let θ, θ Θ be distinct parameter vectors and let N be a positive integer. Let Γ be as given by Deﬁnition 3. { ∈ } { By the assumption of signiﬁcant correlation contribution, the ﬁnite sequences c (τ; θ) : τ Γ and c (τ; θ) : ∈ } (cid:101) (cid:101) ∈ X X τ Γ are not equal. Since c(N)(τ; θ) = c(N)(τ)c (τ; θ) for τ Γ, and according to (10), for N large enough the { (cid:101) ∈ X} { (cid:101) g X ∈ (cid:101)} sequences c(N)(τ; θ) : τ Γ and c(N)(τ; θ) : τ Γ are not equal. Hence for N large enough the sequences { (cid:101) X − − · · · − X} { (cid:101) (cid:101) − − · · · − } c(N)(τ; θ) : τ = (N 1), , N 1) and c(N)(τ; θ) : τ = (N 1), , N 1) are not equal. Their ﬁnite X { (cid:101) ∈ } {X (cid:101) ∈ } (N) (N) Fourier transforms S (ω; θ) : ω Ω and S (ω; θ) : ω Ω , are by the bijective nature of the Fourier X N X N transform, not equal either. (cid:48) ∈ This means that for two distinct parameters vectors θ, θ Θ, we will have two distinct expected periodograms. This is a necessary condition for an estimation procedure based on the expected periodogram. We will propose such an estimation procedure in Section , and derive its consistency and convergence rate in Section . 2.4 Bivariate modulated processes It is common in pratical applications to observe more than one time series at any time, and to analyse a set together. Often the series in the set are related via phase-shifts and other small temporal inhomogeneities, see e.g. Allen and Robertson (1996), Ru¨nstler (2004), Allefeld et al. (2009), Lilly and Olhede (2012). Bivariate nonstationary processes can be challenging to model, as they may not be representable in the same nonstationary oscillatory family (Tong, 1973, 1974). To explore the nature of multivariate modulation, we shall investigate the representation of bivariate processes. For ease of exposition we shall represent such series using complex- valued time series, see Walker (1993). We shall continue to assume that the latent process, now denoted Z for t complex-valued processes, is Gaussian and zero-mean, leaving only the second order structure to be modelled. { ∗ } For complex-valued processes both the autocovariance c (τ) = E Z Z (the star denotes conjugation) and the { } Z t t+τ relation r (τ) = E Z Z sequences need to be modelled (Walden, 2013). Complex-valued processes, unlike real- Z t t+τ valued, no longer have a spectrum that needs to satisfy Hermitian symmetry, and if the series represents motion in the plane, the positive and negative frequencies represent clockwise and anti-clockwise rotations respectively. Following the classical modelling framework (Miller, 1969) for complex-valued processes we shall assume that the relation sequence takes the value zero for all lags. The complex-valued process is then said to be proper. The assumption of propriety has the consequence of directly extending equation (1) to the complex-valued case from the real-valued case. Speciﬁcally, let Z be a complex-valued Gaussian proper zero-mean process, a complex-valued t modulated process is deﬁned as one taking the form(cid:101), Z = g Z , (20) t t t ∈ N at all times t , where gt = ρteiφt is a bounded modulation sequence. We note that for complex-valued time series the modulation sequence is complex-valued. With this deﬁ(cid:101)nition, the modulation series g accomplishes a t time-dependent rescaling or expansion/dilation, from ρt, together with a time-dependent rotation, from eiφt. The autocovariance of the co(cid:110)mplex-valu(cid:111)ed modulated process Zt at times t1 and t2 is given by the conveniently simple form, (cid:101) (cid:101) (cid:101) ∗ ∗ − − − cZ(t1, t2; θ) = E Zt1Zt2; θ = gt1gt2cZ(t2 t1; θ) = ρt1ρt2ei(φt2 φt1)cZ(t2 t1; θ), (cid:101) − and c (t , t ; θ) fully characterizes the process. Note that this quantity is not only a function of the lag t t Z 1 2 2 1 as the process is no longer stationary. Similarly to the univariate case cf. (4), let N be any positive integer, we · · · − deﬁne for τ = 0, , N 1, (cid:88) − − N τ 1 1 ∗ c(N)(τ) = g g . (21) g N t t+τ t=0 (cid:110) (cid:111) Note that when g is real-valued (21) and (4) are the same. We also extend the notion of a signiﬁcant correlation t contribution for complex-valued modulated processes, which nat(cid:101)urally mim(cid:101)ics D(cid:101) eﬁ(cid:101)nition 3. We deﬁne the expected periodogram of a complex-valued modulated time series as S(N)(ω) = E Sˆ(N)(ω); θ , which can be computed Z Z eﬃciently similarly to Proposition 2 for the univariate case, by replacing X by Z in (3) and (19). t t A univariate real-valued modulated process is stationary if and only if the modulating sequence is a constant. A necessary and suﬃcient condition on the modulating sequence for the complex-valued modulated process (20) to be stationary is more complicated to obtain, and is determined in the following proposition. 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Proof.

text: Let

text: θ,

text: θ

text: Θ

text: be

text: distinct

text: parameter

text: vectors

text: and

text: let

text: N

text: be

text: a

text: positive

text: integer.

text: Let

text: Γ

text: be

text: as

text: given

text: by

text: Deﬁnition

text: By

text: the

text: assumption

text: of

text: signiﬁcant

text: correlation

text: contribution,

text: the

text: ﬁnite

text: sequences

text: c

text: (τ;

text: θ)

text: τ

text: Γ

text: and

text: c

text: (τ;

text: θ)

text: X

text: X

text: τ

text: Γ

text: are

text: not

text: equal.

text: Since

text: c(N)(τ;

text: θ)

text: c(N)(τ)c

text: (τ;

text: θ)

text: for

text: τ

text: Γ,

text: and

text: according

text: to

text: for

text: N

text: large

text: enough

text: the

text: X}

text: g

text: X

text: sequences

text: c(N)(τ;

text: θ)

text: τ

text: Γ

text: and

text: c(N)(τ;

text: θ)

text: τ

text: Γ

text: are

text: not

text: equal.

text: Hence

text: for

text: N

text: large

text: enough

text: the

text: sequences

text: X

text: X}

text: c(N)(τ;

text: θ)

text: τ

text: (N

text: N

text: and

text: c(N)(τ;

text: θ)

text: τ

text: (N

text: N

text: are

text: not

text: equal.

text: Their

text: ﬁnite

text: X

text: {X

text: (N)

text: (N)

text: Fourier

text: transforms

text: S

text: (ω;

text: θ)

text: ω

text: Ω

text: and

text: S

text: (ω;

text: θ)

text: ω

text: Ω

text: are

text: by

text: the

text: bijective

text: nature

text: of

text: the

text: Fourier

text: X

text: N

text: X

text: N

text: transform,

text: not

text: equal

text: either.

text: This

text: means

text: that

text: for

text: two

text: distinct

text: parameters

text: vectors

text: θ,

text: θ

text: Θ,

text: we

text: will

text: have

text: two

text: distinct

text: expected

text: periodograms.

text: This

text: is

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text: PSDid 102 0 linear FE 10 101 cPuUbFicE FE Res1 100 Res2 Res 10−1 3 10−2 10 -5 Res4 10−3 10−4 10−5 -10 10 10−6 8 16 32 64 128 256 512 0 5 10 15 20 Number of elements eigen iteration Figure 5.1: Left: error of the sum of four smallest eigenvalues of (H, S) with respect to the number of FEs of three diﬀerent FE discretizatoins in Example 5.1. Right: convergence of the PSD-id method for computing four smallest eigenvalues. and τ = a(0.01 0.02 0.03), τ = a(0.51 0.47 0.55), Ce Al with lattice parameter a = 5.75 bohr. Since Ce has a full complement of s, p, d, and f states in valence, it requires 17 enrichment functions to span the occupied space. The near-dependencies between the ﬁnite element basis functions and the enriched basis functions lead to an ill-conditioned generalized eigenvalue problem (1.1). × In this numerical example, the matrix size of H and S is n = 7 83 + 1752 = 5336. Both H × × and S are ill conditioned and their condition numbers are (κ (H), κ (S)) = (1.1641 1010, 2.5731 2 2 1011). Furthermore, H and S share a common near-nullspace span(V ) of dimension 1000 such that (cid:107) (cid:107) (cid:107) (cid:107) − HV = SV = O(10 4), where V is orthonormal. This is an extremely ill-conditioned eigenvalue problem. Figure 5.2 shows the convergence history of the PSD-id method for c(cid:101)omputing four smallest eigenvalues. As in Figure 5.1, the localization of the j approximate eigenpa(cid:101)ir (λi;j, ui;−j) is ma−rked by “+” sign. Once λ is localized, the locally accelerated preconditioner K = (H λ S) 1 is i;j i;j i;j used. Again, as Theorem 4.1 predicts, the locally accelerated preconditioner K leads to superlinear i;j convergence of the PSD-id algorithm. 6 Conclusion In this paper, we proved the convergence of the PSD-id method, and derived a nonasymptotic estimate of the rate of convergence of the method. We show that with the proper choice of the shift, the indeﬁnite shift-and-invert preconditioner is a locally accelerated preconditioner and leads to superlinear convergence. Two numerical examples are presented to verify the theoretical results on the convergence behavior of the PSD-id method for solving ill-conditioned Hermitian-deﬁnite generalized eigenvalue problems. References [1] J. H. Bramble, J. E. Pasciak, and A. V. Knyazev. A subspace preconditioning algorithm for eigenvector/eigenvalue computation. Adv. Comput. Math., 6(1):159–189, 1996. [2] B. R. Brooks, D. Janeˇziˇc, and M. Karplus. Harmonic analysis of large systems. I. methodology. J. Comput. Chem., 16(12):1522–1542, 1995. 19 seulavnegie tsellams ruof eht fo mus eht ni rorrE laudiser evitaler ##LTFigure## ##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## ##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## ##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## ##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## ##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## ##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## ##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## ##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: PSDid

font: default

text: linear

text: FE

text: cPuUbFicE

text: FE

text: Res

text: Number

text: of

text: elements

text: eigen

text: iteration

text: Figure

text: Left:

label: caption

text: error

text: of

text: the

text: sum

text: of

text: four

text: smallest

text: eigenvalues

text: of

text: (H,

text: S)

text: with

text: respect

text: to

text: the

text: number

text: of

text: FEs

text: of

text: three

text: diﬀerent

text: FE

text: discretizatoins

text: in

text: Example

text: Right:

text: convergence

text: of

text: the

text: PSD-id

text: method

text: for

text: computing

text: four

text: smallest

text: eigenvalues.

text: and

text: τ

text: τ

text: Ce

text: Al

text: with

text: lattice

text: parameter

text: a

text: bohr.

text: Since

text: Ce

text: has

text: a

text: full

text: complement

text: of

text: s,

text: p,

text: d,

text: and

text: f

text: states

text: in

text: valence,

text: it

text: requires

text: enrichment

text: functions

text: to

text: span

text: the

text: occupied

text: space.

text: The

text: near-dependencies

text: between

text: the

text: ﬁnite

text: element

text: basis

text: functions

text: and

text: the

text: enriched

text: basis

text: functions

text: lead

text: to

text: an

text: ill-conditioned

text: generalized

text: eigenvalue

text: problem

text: In

text: this

text: numerical

text: example,

text: the

text: matrix

text: size

text: of

text: H

text: and

text: S

text: is

text: n

text: Both

text: H

text: and

text: S

text: are

text: ill

text: conditioned

text: and

text: their

text: condition

text: numbers

text: are

text: (κ

text: (H),

text: κ

text: (S))

text: Furthermore,

text: H

text: and

text: S

text: share

text: a

text: common

text: near-nullspace

text: span(V

text: of

text: dimension

text: such

text: that

text: HV

text: SV

text: where

text: V

text: is

text: orthonormal.

text: This

text: is

text: an

text: extremely

text: ill-conditioned

text: eigenvalue

text: problem.

text: Figure

text: shows

text: the

text: convergence

text: history

text: of

text: the

text: PSD-id

text: method

text: for

text: four

text: smallest

text: eigenvalues.

text: As

text: in

text: Figure

text: the

text: localization

text: of

text: the

text: j

text: approximate

text: (λi;j,

text: ui;−j)

text: is

text: ma−rked

text: by

text: sign.

text: Once

text: λ

text: is

text: localized,

text: the

text: locally

text: accelerated

text: preconditioner

text: K

text: (H

text: λ

text: S)

text: is

text: i;j

text: i;j

text: i;j

text: used.

text: Again,

text: as

text: Theorem

text: predicts,

text: the

text: locally

text: accelerated

text: preconditioner

text: K

text: leads

text: to

text: superlinear

text: i;j

text: convergence

text: of

text: the

text: PSD-id

text: algorithm.

text: Conclusion

text: In

text: this

text: paper,

text: we

text: proved

text: the

text: convergence

text: of

text: the

text: PSD-id

text: method,

text: and

text: derived

text: a

text: nonasymptotic

text: estimate

text: of

text: the

text: rate

text: of

text: convergence

text: of

text: the

text: method.

text: We

text: show

text: that

text: with

text: the

text: proper

text: choice

text: of

text: the

text: shift,

text: the

text: indeﬁnite

text: shift-and-invert

text: preconditioner

text: is

text: a

text: locally

text: accelerated

text: preconditioner

text: and

text: leads

text: to

text: superlinear

text: convergence.

text: Two

text: numerical

text: examples

text: are

text: presented

text: to

text: verify

text: the

text: theoretical

text: results

text: on

text: the

text: convergence

text: behavior

text: of

text: the

text: PSD-id

text: method

text: for

text: solving

text: ill-conditioned

text: Hermitian-deﬁnite

text: generalized

text: eigenvalue

text: problems.

text: References

text: J.

text: H.

text: Bramble,

text: J.

text: E.

text: Pasciak,

text: and

text: A.

text: V.

text: Knyazev.

text: A

text: subspace

text: preconditioning

text: algorithm

text: for

text: eigenvector/eigenvalue

text: computation.

text: Adv.

text: Comput.

text: Math.,

text: B.

text: R.

text: Brooks,

text: D.

text: Janeˇziˇc,

text: and

text: M.

text: Karplus.

text: Harmonic

text: analysis

text: of

text: large

text: systems.

text: I.

text: methodology.

text: J.

text: Comput.

text: Chem.,

text: seulavnegie

text: tsellams

text: ruof

text: eht

text: fo

text: mus

text: eht

text: ni

text: rorrE

text: laudiser

text: evitaler

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text: 11 have been factorized. The H(1)(z) are Hankel functions of the ﬁrst kind and describe the outgoing waves. The (cid:96) relative weight of upper and lower Dirac components − ≡ k (cid:15) m κ = , (A4) (cid:15) + m k has been introduced to make Hermiticity in the coupled equations explicit, see below. It is convenient to deﬁne × (cid:32) (cid:33) (cid:32) (cid:33) 2 2 submatrices − − 1 0 ic s − † H = αH (cid:32) , P (cid:33)= αP (cid:32) 2 2 = P , (cid:33) (cid:32) (cid:33) 0 1 s ic 2 2 − − − − s s c + ic s s s c ic s ic s G = α −2 ∆ ∆− 2 ∆ + α 2 ξ ξ 2 ξ + α 2 2 (A5) G ξ r c ic s s s c + ic s s s s ic ∆ 2 ∆ 2 ∆ ξ 2 ξ 2 ξ 2 2 † Note that for c = 0 and s = 1 these are the matrices as deﬁned in eq. (B3) of ref.[18] with G = G and G− = G . 2 2 + (cid:32) (cid:33) (cid:32) (cid:33) (cid:32) (cid:33) (cid:32) (cid:33) With these deﬁnitions the potential matrices become very compact: − † − † H G H G − G P G P † Vuu = G† H , Vdd = G −H , Vud = P G† , Vdu = P −G = Vud . (A6) Even though the problem is manifestly Hermitian, the matrix elements are no longer real. The diﬀerential equations for outgo(cid:2)ing bounda(cid:3)ry conditions are als(cid:2)o discussed(cid:3) in appendix B of ref.[18] F (cid:2)M (cid:3) · F F · M (cid:2) M (cid:3) · G · ∂ = + O + (r) + k + C Z ρ ff d ff fg d G M · G G · M M − · F · ∂ = + O + (r) + k C Z , (A7) ρ gg u gg gf u × where the 4 (cid:32)4 coeﬃcient matrices without an overline ar(cid:33)e purely kinematic, H(1) (kρ) H(1)(kρ) H(1) (kρ) H(1) (kρ) − Z = diag (cid:96)+1 , (cid:96) , (cid:96)+2 , (cid:96)+1 , Z = (Z ) 1 , (A8) u H(1) (kρ) H(1) (kρ) H(1) (kρ) H(1)(kρ) d u (cid:96)+2 (cid:96)+1 (cid:96)+1 (cid:96) 1 − − 1 − − O = diag ( ((cid:96) + 2), ((cid:96) + 1), (cid:96) + 1, (cid:96)) , O = diag ((cid:96) + 1, (cid:96), ((cid:96) + 2), ((cid:96) + 1)) u ρ d ρ − − F G and C = diag( 1, 1, 1, 1). The matrices multiplying and from the right are also independent of the background potential, M M − · − M M · − (r) = (r)(k) = kC Z (k) O and (r) = (r)(k) = kC Z (k) O . (A9) ff ff d d gg gg u u Genuine interactions from the string background are solely contained in the overlined matrices in Eq. (A7). Using × (cid:32) (cid:33) (cid:32) (cid:33) the same 2 2 matrix notation as above, we have explicitly − † M G P M − G P gg = CVud = − − †(cid:32) (cid:33) ff = CVdu = − (cid:32) (cid:33) P G P G (A10) − − − † M H G M − H G gf = E−1m CVuu = E−1m G† H fg = E+1m CVdd = E+1m −G H . ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: have

text: been

text: factorized.

text: The

text: are

text: Hankel

text: functions

text: of

text: the

text: ﬁrst

text: kind

text: and

text: describe

text: the

text: outgoing

text: waves.

text: The

text: relative

text: weight

text: of

text: upper

text: and

text: lower

text: Dirac

text: components

text: k

text: m

text: κ

text: m

text: k

text: has

text: been

text: introduced

text: to

text: make

text: Hermiticity

text: in

text: the

text: coupled

text: equations

text: explicit,

text: see

text: below.

text: It

text: is

text: convenient

text: to

text: deﬁne

text: submatrices

text: ic

text: s

text: H

text: αH

text: P

text: αP

text: P

text: s

text: ic

text: s

text: s

text: c

text: ic

text: s

text: s

text: s

text: c

text: ic

text: s

text: ic

text: s

text: G

text: α

text: −2

text: α

text: ξ

text: ξ

text: ξ

text: α

text: G

text: ξ

text: r

text: c

text: ic

text: s

text: s

text: s

text: c

text: ic

text: s

text: s

text: s

text: s

text: ic

text: ξ

text: ξ

text: ξ

text: Note

text: that

text: for

text: c

text: and

text: s

text: these

text: are

text: the

text: matrices

text: as

text: deﬁned

text: in

text: eq.

text: of

text: with

text: G

text: G

text: and

text: G−

text: G

text: With

text: these

text: deﬁnitions

text: the

text: potential

text: matrices

text: become

text: very

text: compact:

text: H

text: G

text: H

text: G

text: G

text: P

text: G

text: P

text: Vuu

text: G†

text: H

text: Vdd

text: G

text: −H

text: Vud

text: P

text: G†

text: Vdu

text: P

text: −G

text: Vud

text: Even

text: though

text: the

text: problem

text: is

text: manifestly

text: Hermitian,

text: the

text: matrix

text: elements

text: are

text: no

text: longer

text: real.

text: The

text: diﬀerential

text: equations

text: for

text: conditions

text: are

text: in

text: appendix

text: B

text: of

text: F

text: F

text: F

text: M

text: M

text: G

text: O

text: (r)

text: k

text: C

text: Z

text: ρ

text: ff

text: d

text: ff

text: fg

text: d

text: G

text: M

text: G

text: G

text: M

text: M

text: F

text: O

text: (r)

text: k

text: C

text: Z

text: ρ

text: gg

text: u

text: gg

text: gf

text: u

text: where

text: the

text: coeﬃcient

text: matrices

text: without

text: an

text: overline

text: purely

text: kinematic,

text: (kρ)

text: (kρ)

text: (kρ)

text: Z

text: diag

text: Z

text: (Z

text: u

text: (kρ)

text: (kρ)

text: (kρ)

text: d

text: u

text: O

text: diag

text: O

text: diag

text: u

text: ρ

text: d

text: ρ

text: F

text: G

text: and

text: C

text: diag(

text: The

text: matrices

text: multiplying

text: and

text: from

text: the

text: right

text: are

text: also

text: independent

text: of

text: the

text: background

text: potential,

text: M

text: M

text: M

text: M

text: (r)

text: (r)(k)

text: kC

text: Z

text: (k)

text: O

text: and

text: (r)

text: (r)(k)

text: kC

text: Z

text: (k)

text: O

text: ff

text: ff

text: d

text: d

text: gg

text: gg

text: u

text: u

text: Genuine

text: interactions

text: from

text: the

text: string

text: background

text: are

text: solely

text: contained

text: in

text: the

text: overlined

text: matrices

text: in

text: Eq.

text: Using

text: the

text: same

text: matrix

text: notation

text: as

text: above,

text: we

text: have

text: explicitly

text: M

text: G

text: P

text: M

text: G

text: P

text: gg

text: CVud

text: ff

text: CVdu

text: P

text: G

text: P

text: G

text: M

text: H

text: G

text: M

text: H

text: G

text: gf

text: CVuu

text: G†

text: H

text: fg

text: CVdd

text: −G

text: H

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 14 [14] N. Graham, M. Quandt, H. Weigel, Lect. Notes Phys. 777 (2009) 1. [15] H. Weigel, M. Quandt, N. Graham, Mod. Phys. Lett. A 30 (2015) 1530022 . [16] H. Weigel, M. Quandt, Phys. Lett. B 690 (2010) 514. [17] H. Weigel, M. Quandt, N. Graham, O. Schro¨der, Nucl. Phys. B 831 (2010) 306. [18] N. Graham, M. Quandt, H. Weigel, Phys. Rev. D 84 (2011) 025017. [19] E. Farhi, N. Graham, R. L. Jaﬀe, H. Weigel, Nucl. Phys. B 630 (2002) 241 . [20] F. R. Klinkhamer, C. Rupp, Nucl. Phys. B 495 (1997) 172 . [21] N. Graham, R. L. Jaﬀe, V. Khemani, M. Quandt, O. Schro¨der, H. Weigel, Nucl. Phys. B 677 (2004) 379. [22] E. Farhi, N. Graham, R. L. Jaﬀe, H. Weigel, Nucl. Phys. B 585 (2000) 443. [23] E. Farhi, N. Graham, R. L. Jaﬀe, H. Weigel, Nucl. Phys. B 595 (2001) 536. [24] P. Pasipoularides, Phys. Rev. D 64 (2001) 105011, hep-th/0502238. [25] O. Schro¨der, N. Graham, M. Quandt, H. Weigel, J. Phys. A 41 (2008) 164049. [26] F. R. Klinkhamer, P. Olesen, Nucl. Phys. B 422 (1994) 227 . [27] N. Graham, M. Quandt, O. Schr¨oder, H. Weigel, Nucl. Phys. B 758 (2006) 112 . [28] J. S. Faulkner, J. Phys. C:. Solid State Phys.,10 (1977) 4661. [29] N. Graham, R. L. Jaﬀe, M. Quandt, H. Weigel, Annals Phys. 293 (2001) 240. [30] N. Graham, R. L. Jaﬀe, M. Quandt, H. Weigel, Phys. Rev. Lett. 87 (2001) 131601. [31] R. G. Newton, Scattering Theory of Waves and Particles Springer, New York (1982). [32] K. Chadan, P. Sabatier, Inverse Problems in Quantum Scattering Theory Springer, New York (1977). [33] G. Ripka, S. Kahana, Phys. Rev. D 36 (1987) 1233. [34] J. Hartmann, F. Beck, W. Bentz, Phys. Rev. C 50 (1994) 3088. [35] E. Anderson el al., “LAPACK Users’ Guide” (1999), Soc. for Industrial & Applied Mathematics, ISBN 0-89871-447-8. [36] D. Diakonov, M. V. Polyakov, P. Sieber, J. Schaldach, K. Goeke, Phys. Rev. D 49 (1994) 6864. [37] H. Weigel, M. Quandt, N. Graham, Phys. Rev. Lett. 106 (2011) 101601. [38] M. Abramowitz, I. Stegun (eds.), Handbook of mathematical functions, Dover, New York (1968).

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text: New

text: York


text: REFERENCES 8 × − × − happens in the electron density range from 1 1017 cm 3 to 5 1018 cm 3 for doped × − ZnO (Fig. 4, symbols). The velocity stops decreasing at around 5 1018 cm 3, and the local maximum forms at higher densities at intermediate electric ﬁelds in the range from 20 kV/cm to 60 kV/cm (triangles, squares, stars). The thermal runaway makes it impossible to resolve the maximum at electric ﬁelds of 80 kV/cm and 100 kV/cm − (diamonds, circles). The maximum is located at above 1019 cm 3. Its position (Fig. 4) correlates with that of the resonance decay of hot phonons [8]. Thus, the observed fast decay of hot phonons supports high values of hot-electron drift velocity unless the thermal runaway comes into play. 5. Conclusions The nanosecond pulsed measurements yielded hot-electron drift velocity in agreement with those of the Monte Carlo simulation when the density of the ionized impurities is allowed to exceed that of the electrons by 10-13 times in bulk ZnO. The Ohm law approximately held in the electric ﬁeld range up to 100 kV/cm at an electron × − density of 1.42 1017 cm 3 but the range shrank at higher electron densities. The LO- phonon–plasmon resonance supported increase in the electron drift velocity is resolved ∼ − − at ( 1019 cm 3). The non-ohmic behavior at higher densities (>1019 cm 3) is caused by increase in the background temperature represented by acoustic phonons. A drift ∼ × velocity of 1.5 107 cm/s was measured at 104 kV/cm ﬁeld. A lot of eﬀort still to be done to design ZnO-based ﬁeld-eﬀect transistors with improved electron transport along a channel. Acknowledgement This research is funded by the Research Council of Lithuania (grant No. APP- 5/2016). References [1] Morko¸c H 2009, Zinc Oxide: Fundamentals, Materials and Device Technology (Weinheim: Wiley-VCH) ¨ ¨ [2] Liu H, Avrutin V, Izyumskaya N, Ozgu¨r U, Morko¸c H 2010 Supperlattices Microstruct. 48 458 [3] Look D C, Leedy K D, Tomich D H, and Bayraktaroglu B 2006 Appl. Phys. Lett. 96 062102 ¨ ¨ [4] Liu Y, Avrutin V, Izyumskaya N, Ozgu¨r U, Yankovich A B, Kvit A V, Voyles P M, and Morko¸c H 2012 J. Appl. Phys 111 103713 [5] Tsukazaki A, Ohtomo A, and Kawasaki M 2006 Appl. Phys. Lett. 88 152106 ˇ ˇ [6] Matulionis A, Liberis J, Sermukˇsnis E, Ardaraviˇcius L, Simukoviˇc A, Kayis C, Zhu ¨ ¨ C Y, Ferreyra R, Avrutin V, Ozgu¨r U, Morkoc¸ H 2012 Microelectron. Reliab. 52 2149 [7] Matulionis A, Liberis J, Matulioniene˙ I, Ramonas M, and 2010 Proc. IEEE 98 1118

text: REFERENCES

text: happens

text: in

text: the

text: electron

text: density

text: range

text: from

text: cm

text: to

text: cm

text: for

text: doped

text: ZnO

text: (Fig.

text: symbols).

text: The

text: velocity

text: stops

text: decreasing

text: at

text: around

text: cm

text: and

text: the

text: local

text: maximum

text: forms

text: at

text: higher

text: densities

text: at

text: intermediate

text: electric

text: ﬁelds

text: in

text: the

text: range

text: from

text: kV/cm

text: to

text: kV/cm

text: (triangles,

text: squares,

text: stars).

text: The

text: thermal

text: runaway

text: makes

text: it

text: impossible

text: to

text: resolve

text: the

text: maximum

text: at

text: electric

text: ﬁelds

text: of

text: kV/cm

text: and

text: kV/cm

text: (diamonds,

text: circles).

text: The

text: maximum

text: is

text: located

text: at

text: above

text: cm

text: Its

text: position

text: (Fig.

text: correlates

text: with

text: that

text: of

text: the

text: resonance

text: decay

text: of

text: hot

text: phonons

text: Thus,

text: the

text: observed

text: fast

text: decay

text: of

text: hot

text: phonons

text: supports

text: high

text: values

text: of

text: hot-electron

text: drift

text: velocity

text: unless

text: the

text: thermal

text: runaway

text: comes

text: into

text: play.

text: Conclusions

text: The

text: nanosecond

text: pulsed

text: measurements

text: yielded

text: hot-electron

text: drift

text: velocity

text: in

text: agreement

text: with

text: those

text: of

text: the

text: Monte

text: Carlo

text: simulation

text: when

text: the

text: density

text: of

text: the

text: ionized

text: impurities

text: is

text: allowed

text: to

text: exceed

text: that

text: of

text: the

text: electrons

text: by

text: times

text: in

text: bulk

text: ZnO.

text: The

text: Ohm

text: law

text: approximately

text: held

text: in

text: the

text: electric

text: ﬁeld

text: range

text: up

text: to

text: kV/cm

text: at

text: an

text: electron

text: density

text: of

text: cm

text: but

text: the

text: range

text: shrank

text: at

text: higher

text: electron

text: densities.

text: The

text: LO-

text: phonon–plasmon

text: resonance

text: supported

text: increase

text: in

text: the

text: electron

text: drift

text: velocity

text: is

text: resolved

text: at

text: cm

text: The

text: non-ohmic

text: behavior

text: at

text: higher

text: densities

text: cm

text: is

text: caused

text: by

text: increase

text: in

text: the

text: background

text: temperature

text: represented

text: by

text: acoustic

text: phonons.

text: A

text: drift

text: velocity

text: of

text: cm/s

text: was

text: measured

text: at

text: kV/cm

text: ﬁeld.

text: A

text: lot

text: of

text: eﬀort

text: still

text: to

text: be

text: done

text: to

text: design

text: ZnO-based

text: ﬁeld-eﬀect

text: transistors

text: with

text: improved

text: electron

text: transport

text: along

text: a

text: channel.

text: Acknowledgement

text: This

text: research

text: is

text: funded

text: by

text: the

text: Research

text: Council

text: of

text: Lithuania

text: (grant

text: No.

text: APP-

text: References

text: Morko¸c

text: H

text: Zinc

text: Oxide:

text: Fundamentals,

text: Materials

text: and

text: Device

text: Technology

text: (Weinheim:

text: Wiley-VCH)

text: Liu

text: H,

text: Avrutin

text: V,

text: Izyumskaya

text: N,

text: Ozgu¨r

text: U,

text: Morko¸c

text: H

text: Supperlattices

text: Microstruct.

text: Look

text: D

text: C,

text: Leedy

text: K

text: D,

text: Tomich

text: D

text: H,

text: and

text: Bayraktaroglu

text: B

text: Appl.

text: Phys.

text: Lett.

text: Liu

text: Y,

text: Avrutin

text: V,

text: Izyumskaya

text: N,

text: Ozgu¨r

text: U,

text: Yankovich

text: A

text: B,

text: Kvit

text: A

text: V,

text: Voyles

text: P

text: M,

text: and

text: Morko¸c

text: H

text: J.

text: Appl.

text: Phys

text: Tsukazaki

text: A,

text: Ohtomo

text: A,

text: and

text: Kawasaki

text: M

text: Appl.

text: Phys.

text: Lett.

text: ˇ

text: ˇ

text: Matulionis

text: A,

text: Liberis

text: J,

text: Sermukˇsnis

text: E,

text: Ardaraviˇcius

text: L,

text: Simukoviˇc

text: A,

text: Kayis

text: C,

text: Zhu

text: C

text: Y,

text: Ferreyra

text: R,

text: Avrutin

text: V,

text: Ozgu¨r

text: U,

text: Morkoc¸

text: H

text: Microelectron.

text: Reliab.

text: Matulionis

text: A,

text: Liberis

text: J,

text: Matulioniene˙

text: I,

text: Ramonas

text: M,

text: and

text: Proc.

text: IEEE


text: REFERENCES 9 ˇ [8] Sermukˇsnis E, Liberis J, Ramonas M, Matulionis A, Toporkov M, Liu H Y, Avrutin ¨ ¨ V, Ozgu¨r U, and Morko¸c H 2015 J. Appl. Phys. 117 065704 ˇ [9] Liberis J, Ramonas M, Sermukˇsnis E, Sakalas P, Szabo N, Schuster M, Wachowiak A, and Matulionis A 2014 Semicond. Sci. Technol. 29 045018 [10] Ardaraviˇcius L, Liberis J, Kiprijanoviˇc O, Matulionis A, Wu M, and Morko¸c H 2011 Phys. Status Solidi RRL 5 65 [11] Khurgin J, Ding Y J, and Jena D 2007 Appl. Phys. Lett. 91 252104 [12] Dyson A and Ridley B K 2010 J. Appl. Phys. 108 104504 [13] Sasa S, Hayafuji T, Kawasaki M, Nakashima A, Koike K, Yano M, and Inoue M 2008 Phys. Status Solidi C 5 115 [14] Sasa S, Maitani T, Furuya Y, Amano T, Koike K, Yano M, and Inoue M 2011 Phys. Status A 208 449 [15] Ardaraviˇcius L, Ramonas M, Liberis J, Kiprijanoviˇc O, Matulionis A, Xie J, Wu M, Leach J H, and Morko¸c H 2009 J. Appl. Phys. 106 073708 ˇ [16] Ardaraviˇcius L, Kiprijanoviˇc O, Liberis J, Matulionis A, Sermukˇsnis E, Ferreyra ¨ ¨ R A, Avrutin V, Ozgu¨r U, and Morko¸c H 2015 Semicond. Sci. Technol. 30 105016 [17] Furno E, Bertazzi F, Goano M, Ghione G, and Bellotti E 2008 Solid-State Electron. 52 1796 [18] Ramonas M, Matulionis A, Liberis J, Eastman L F, Chen X, and Sun Y J 2005 Phys. Rev. B 71 075324 [19] Ridley B K 1996 J. Phys.: Condens. Matter 8 L511 [20] Ardaraviˇcius L, Matulionis A, Liberis J, Kiprijanovic O., Ramonas M, Eastman L F, Shealy J. R, and Vertiatchikh A 2003 Appl. Phys. Lett. 83 4038 [21] Matulionis A 2013 Semicond. Sci. Technol. 28 074007

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text: Toporkov

text: M,

text: Liu

text: H

text: Y,

text: Avrutin

text: V,

text: Ozgu¨r

text: U,

text: and

text: Morko¸c

text: H

text: J.

text: Appl.

text: Phys.

text: ˇ

text: Liberis

text: J,

text: Ramonas

text: M,

text: Sermukˇsnis

text: E,

text: Sakalas

text: P,

text: Szabo

text: N,

text: Schuster

text: M,

text: Wachowiak

text: A,

text: and

text: Matulionis

text: A

text: Semicond.

text: Sci.

text: Technol.

text: Ardaraviˇcius

text: L,

text: Liberis

text: J,

text: Kiprijanoviˇc

text: O,

text: Matulionis

text: A,

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text: M,

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text: H

text: Phys.

text: Status

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text: D

text: Appl.

text: Phys.

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text: Phys.

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text: Status

text: A

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text: L,

text: Ramonas

text: M,

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text: A,

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text: J,

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text: J

text: H,

text: and

text: Morko¸c

text: H

text: J.

text: Appl.

text: Phys.

text: ˇ

text: Ardaraviˇcius

text: L,

text: Kiprijanoviˇc

text: O,

text: Liberis

text: J,

text: Matulionis

text: A,

text: Sermukˇsnis

text: E,

text: Ferreyra

text: R

text: A,

text: Avrutin

text: V,

text: Ozgu¨r

text: U,

text: and

text: Morko¸c

text: H

text: Semicond.

text: Sci.

text: Technol.

text: Furno

text: E,

text: Bertazzi

text: F,

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text: M,

text: Ghione

text: G,

text: and

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text: Phys.:

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text: L,

text: Matulionis

text: A,

text: Liberis

text: J,

text: Kiprijanovic

text: O.,

text: Ramonas

text: M,

text: Eastman

text: L

text: F,

text: Shealy

text: J.

text: R,

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text: A

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text: Phys.

text: Lett.

text: Matulionis

text: A

text: Semicond.

text: Sci.

text: Technol.


text: 4 − ∂S 2S( 4b2π + bR S + 4πΦ2) C = T( ) = 0− . (15) Φ ∂T Φ 4b2π + bR S 4πΦ2 0 Note that the denominator of Eq.(15) is exactly the same as the numerator of Eq.(10), implying that the divergence of C corresponds to the minimum Hawking temperature. Φ − When b = 1, R = 12/l2 = 4Λ, Eq.(15) reduces to 0 − − 2S[π(1 Φ2) ΛS] C = − − − , (16) Φ ΛS π(1 Φ2) reproducing the result of RN-AdS black holes [36]. Substituting Eq.(6) into Eq.(15), one can express the speciﬁc heat into the function of r as + − 2br2 ( 4b2π + bR πr2 + 4πΦ2) C = + 0 −+ . (17) Φ 4b2 + b2R r2 4Φ2 0 + One can also derive the analog of volume expansion coeﬃcient and isothermal compressibility coeﬃcient as − 1 ∂Q 16b2πr α = ( ) = +− , (18) Q ∂T Φ 4b2 + b2R r2 4Φ2 0 + − 1 ∂Q 4b2 + b2R r2 12Φ2 κ = ( ) = 0 + − . (19) T Q ∂Φ T Φ(4b2 + b2R r2 4Φ2) 0 + − Comparing Eqs.(18) and (19) with Eq.(17), one can ﬁnd that both α and κ share the same factor 4b2+b2R r2 4Φ2 T 0 + in their denominators as the speciﬁc heat. It is not diﬃcult to ﬁnd the condition corresponding to the divergence of C , α and κ as Φ T − 4b2 + b2R r2 4Φ2 = 0, (20) 0 + which can be analytically solved as (cid:115) − 2 Φ2 b2 r = . (21) + b R 0 Considering the restrictions b > 0, R < 0, the above root make sense physically only when 0 < Φ < b. 0 Fig. 2(a) shows the case of 0 < Φ < b while Fig. 2(b) shows the case of Φ > b. One can see clearly that the speciﬁc heat C encounters a divergence when 0 < Φ < b while there is no divergence for the case Φ > b. This ﬁnding also Φ diﬀers from our former research in the canonical ensemble [18], where there may be two, one or no divergence points for the speciﬁc heat C . Q Fig. 2(c) and 2(d) show that α and κ diverge at the same place where C does, in accordance with the above T Φ deductions. IV. THERMODYNAMIC GEOMETRY OF f(R) ADS BLACK HOLE To examine the phase structure newly found in Sec. , we would like to appeal to thermodynamic geometry tools, such as Weinhold geometry [22] and Ruppeiner geometry [23], which has found various applications in probing the phase structures of black holes [26]-[40]. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ∂S

text: bR

text: S

text: C

text: T(

text: 0−

text: Φ

text: ∂T

text: Φ

text: bR

text: S

text: 4πΦ2

text: Note

text: that

text: the

text: denominator

text: of

text: is

text: exactly

text: the

text: same

text: as

text: the

text: numerator

text: of

text: implying

text: that

text: the

text: divergence

text: of

text: C

text: corresponds

text: to

text: the

text: minimum

text: Hawking

text: temperature.

text: Φ

text: When

text: b

text: R

text: reduces

text: to

text: ΛS]

text: C

text: Φ

text: ΛS

text: reproducing

text: the

text: result

text: of

text: RN-AdS

text: black

text: holes

text: Substituting

text: into

text: one

text: can

text: express

text: the

text: speciﬁc

text: heat

text: into

text: the

text: function

text: of

text: r

text: as

text: bR

text: C

text: Φ

text: One

text: can

text: also

text: derive

text: the

text: analog

text: of

text: volume

text: expansion

text: coeﬃcient

text: and

text: isothermal

text: compressibility

text: coeﬃcient

text: as

text: ∂Q

text: α

text: Q

text: ∂T

text: Φ

text: ∂Q

text: κ

text: T

text: Q

text: ∂Φ

text: T

text: Comparing

text: and

text: with

text: one

text: can

text: ﬁnd

text: that

text: both

text: α

text: and

text: κ

text: share

text: the

text: same

text: factor

text: T

text: in

text: their

text: denominators

text: as

text: the

text: speciﬁc

text: heat.

text: It

text: is

text: not

text: diﬃcult

text: to

text: ﬁnd

text: the

text: condition

text: corresponding

text: to

text: the

text: divergence

text: of

text: C

text: α

text: and

text: κ

text: as

text: Φ

text: T

text: which

text: can

text: be

text: analytically

text: solved

text: as

text: Φ2

text: r

text: b

text: R

text: Considering

text: the

text: restrictions

text: b

text: R

text: the

text: above

text: root

text: make

text: sense

text: physically

text: only

text: when

text: Φ

text: b.

text: Fig.

text: shows

text: the

text: case

text: of

text: Φ

text: b

text: while

text: Fig.

text: shows

text: the

text: case

text: of

text: Φ

text: b.

text: One

text: can

text: see

text: clearly

text: that

text: the

text: speciﬁc

text: heat

text: C

text: encounters

text: a

text: divergence

text: when

text: Φ

text: b

text: while

text: there

text: is

text: no

text: divergence

text: for

text: the

text: case

text: Φ

text: b.

text: This

text: ﬁnding

text: also

text: Φ

text: diﬀers

text: from

text: our

text: former

text: research

text: in

text: the

text: canonical

text: ensemble

text: where

text: there

text: may

text: be

text: two,

text: one

text: or

text: no

text: divergence

text: points

text: for

text: the

text: speciﬁc

text: heat

text: C

text: Q

text: Fig.

text: and

text: show

text: that

text: α

text: and

text: κ

text: diverge

text: at

text: the

text: same

text: place

text: where

text: C

text: does,

text: in

text: accordance

text: with

text: the

text: above

text: T

text: Φ

text: deductions.

text: IV.

text: THERMODYNAMIC

text: GEOMETRY

text: OF

text: f(R)

text: ADS

text: BLACK

text: HOLE

text: To

text: examine

text: the

text: phase

text: structure

text: newly

text: found

text: in

text: Sec.

text: we

text: would

text: like

text: to

text: appeal

text: to

text: thermodynamic

text: geometry

text: tools,

text: such

text: as

text: Weinhold

text: geometry

text: and

text: Ruppeiner

text: geometry

text: which

text: has

text: found

text: various

text: applications

text: in

text: probing

text: the

text: phase

text: structures

text: of

text: black

text: holes

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##