text: A PERSISTENCE LANDSCAPES TOOLBOX FOR TOPOLOGICAL STATISTICS 15 9 20000 9 900 9 180 8 18000 8 800 8 160 7 16000 7 700 7 140 6 14000 6 600 6 120 5 12000 5 500 5 100 4 10000 4 4 3 8000 3 400 3 80 2 6000 2 300 2 60 1 4000 1 200 1 40 0 2000 0 100 0 20 -1 0 -1 0 -1 0 -1 0 1 2 3 4 5 6 7 8 9 -1 0 1 2 3 4 5 6 7 8 9 -1 0 1 2 3 4 5 6 7 8 9 9 450 9 45 9 14 8 400 8 40 8 12 7 350 7 35 7 6 300 6 30 6 10 5 250 5 25 5 8 4 4 4 3 200 3 20 3 6 2 150 2 15 2 4 1 100 1 10 1 0 50 0 5 0 2 -1 0 -1 0 -1 0 -1 0 1 2 3 4 5 6 7 8 9 -1 0 1 2 3 4 5 6 7 8 9 -1 0 1 2 3 4 5 6 7 8 9 9 18 9 8 9 3.5 8 16 8 7 8 3 567 111024 567 56 567 22.5 4 4 4 4 123 468 123 23 123 11.5 0 2 0 1 0 0.5 -1 0 -1 0 -1 0 -1 0 1 2 3 4 5 6 7 8 9 -1 0 1 2 3 4 5 6 7 8 9 -1 0 1 2 3 4 5 6 7 8 9 ∞ Figure 6. Color plots of L1 (top), L2 (middle) and L (bottom) distance ma- trices for average persistence landscapes for points sampled from spheres in di- { } mensions 2, . . . , 10 normalized so that average distance between points is 1, for homological degree 0 (left), 1 (middle) and 2 (right). 10 bottleneck 10000 W1 10000 W2 L infty L1 L2 1000 1000 1 100 100 10 10 0.1 1 1 0.01 0.1 0.1 0.01 0.01 0.001 0.001 0.001 100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800 Figure 7. Comparison of time of distance computations: (a) Bottleneck versus ∞ L , (b) W1 versus L1, (c) W2 versus L2. Note the logarithmic scale on the vertical axis. Our toy test example for these programs is a set of ﬁles consisting of birth-death pairs, which we ∈ { } now describe. For n 1, 2, . . . , 5 , let A denote the union of the circles of radius one centered at n (0, 0), (2, 0), . . . , (2n, 0). From A we sampled 50n points independently using the uniform measure. n − To each of these points we added a uniform error sampled from [ 0.15, 0.15]2, see Figure 9 for example. From this set of points, we calculated the persistent homology in degrees zero and one of the corresponding Vietoris-Rips complex using Perseus [35]. This was repeated 11 times for each n. The results are encoded in 5 directories (named 1circle,. . . ,5circle) each of which contains × 2 11 = 22 ﬁles that each encode a persistence diagram of either degree zero or one. For the further use in the programs the ﬁles containing homology in a particular degree are listed in a ﬁles 1circle_dim1.txt,...,5circle_dim1.txt . For example, the ﬁle 1circle_dim1.txt lists the following ﬁles, each of which contains a persistence diagram. ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##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## 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text: A PERSISTENCE LANDSCAPES TOOLBOX FOR TOPOLOGICAL STATISTICS 21 Figure 11. Exact versus grid-based estimate of the persistence landscape. The underlying persistence landscape is given by the black lines. The blue dots in the x-axis indicate the grid points. The corresponding red dots in the plot of the persistence landscape give the values at the grid points. The dashed line is the estimated persistence landscape. The blue regions indicate the intersection of the cones in which the persistence landscape is guaranteed to be located. half the sum of the areas of the rectangles described above. If the grid has size m and there are − K nonzero persistence landscape functions, then this is bounded by K(m 1)δ2 . 4 − Even though the slope of an average landscape may be any real number between 1 and 1, in our experience, the slope is often far from the extreme possible values. Therefore the error bounds above may be large overestimates. 7. Configuration of the library. The Persistence Landscape Toolbox library is a collection of C++ programs. Each ﬁle with the .cpp extension contains a ready-to use program. For a description of its functionality and required parameters, please run the program without any parameters. There is a conﬁgure ﬁle which needs to be present in the folder in which a program is run. This ﬁle contains basic conﬁguration parameters for the library. In this ﬁle one can set up the constant which represents inﬁnity. The library assumes that the input ﬁle consists of a collection of numbers as described in the Section . To encode inﬁnite intervals, the inﬁnities have to be changed to a ”magic number” which is deﬁned in the conﬁgure ﬁle. There one can ﬁnd various options for what can be done with the inﬁnite intervals. This ﬁle also allows one to switch the library from the default exact mode to a grid- based mode. If a grid-based mode is used, the user should set up the parameters of the grid in the conﬁguration ﬁle. 8. Conclusion To conclude, we have provided asymptotically optimal algorithms for computing persistence landscapes, averaging them and calculating distances between (average) persistence landscapes. We have implemented these algorithms and demonstrated how they may be used for hypothesis testing and classiﬁcation. We hope that they will of use to the (topological) data scientist. Furthermore, this is not intended to be the end of the story. In future work, we aim to improve these algorithms and related software and increase their usefulness. Appendix A. Persistence landscapes for barcodes with infinite intervals ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

label: paragraph

text: PERSISTENCE

text: LANDSCAPES

text: TOOLBOX

text: FOR

text: TOPOLOGICAL

text: STATISTICS

text: Figure

text: Exact

label: caption

text: versus

text: grid-based

text: estimate

text: of

text: the

text: persistence

text: landscape.

text: The

text: underlying

text: persistence

text: landscape

text: is

text: given

text: by

text: the

text: black

text: lines.

text: The

text: blue

text: dots

text: in

text: the

text: x-axis

text: indicate

text: the

text: grid

text: points.

text: The

text: corresponding

text: red

text: dots

text: in

text: the

text: plot

text: of

text: the

text: persistence

text: landscape

text: give

text: the

text: values

text: at

text: the

text: grid

text: points.

text: The

text: dashed

text: line

text: is

text: the

text: estimated

text: persistence

text: landscape.

text: The

text: blue

text: regions

text: indicate

text: the

text: intersection

text: of

text: the

text: cones

text: in

text: which

text: the

text: persistence

text: landscape

text: is

text: guaranteed

text: to

text: be

text: located.

text: half

text: the

text: sum

text: of

text: the

text: areas

text: of

text: the

text: rectangles

text: described

text: above.

text: If

text: the

text: grid

text: has

text: size

text: m

text: and

text: there

text: are

text: K

text: nonzero

text: persistence

text: landscape

text: functions,

text: then

text: this

text: is

text: bounded

text: by

text: K(m

text: Even

text: though

text: the

text: slope

text: of

text: an

text: average

text: landscape

text: may

text: be

text: any

text: real

text: number

text: between

text: and

text: in

text: our

text: experience,

text: the

text: slope

text: is

text: often

text: far

text: from

text: the

text: extreme

text: possible

text: values.

text: Therefore

text: the

text: error

text: bounds

text: above

text: may

text: be

text: large

text: overestimates.

text: Configuration

text: of

text: the

text: library.

text: The

text: Persistence

text: Landscape

text: Toolbox

text: library

text: is

text: a

text: collection

text: of

text: C++

text: programs.

text: Each

text: ﬁle

text: with

text: the

text: .cpp

text: extension

text: contains

text: a

text: ready-to

text: use

text: program.

text: For

text: a

text: description

text: of

text: its

text: functionality

text: and

text: required

text: parameters,

text: please

text: run

text: the

text: program

text: without

text: any

text: parameters.

text: There

text: is

text: a

text: conﬁgure

text: ﬁle

text: which

text: needs

text: to

text: be

text: present

text: in

text: the

text: folder

text: in

text: which

text: a

text: program

text: is

text: run.

text: This

text: ﬁle

text: contains

text: basic

text: conﬁguration

text: parameters

text: for

text: the

text: library.

text: In

text: this

text: ﬁle

text: one

text: can

text: set

text: up

text: the

text: constant

text: which

text: represents

text: inﬁnity.

text: The

text: library

text: assumes

text: that

text: the

text: input

text: ﬁle

text: consists

text: of

text: a

text: collection

text: of

text: numbers

text: as

text: described

text: in

text: the

text: Section

text: To

text: encode

text: inﬁnite

text: intervals,

text: the

text: inﬁnities

text: have

text: to

text: be

text: changed

text: to

text: a

text: ”magic

text: number”

text: which

text: is

text: deﬁned

text: in

text: the

text: conﬁgure

text: ﬁle.

text: There

text: one

text: can

text: ﬁnd

text: various

text: options

text: for

text: what

text: can

text: be

text: done

text: with

text: the

text: inﬁnite

text: intervals.

text: This

text: ﬁle

text: also

text: allows

text: one

text: to

text: switch

text: the

text: library

text: from

text: the

text: default

text: exact

text: mode

text: to

text: a

text: grid-

text: based

text: mode.

text: If

text: a

text: grid-based

text: mode

text: is

text: used,

text: the

text: user

text: should

text: set

text: up

text: the

text: parameters

text: of

text: the

text: grid

text: in

text: the

text: conﬁguration

text: ﬁle.

text: Conclusion

text: To

text: conclude,

text: we

text: have

text: provided

text: asymptotically

text: optimal

text: algorithms

text: for

text: computing

text: persistence

text: landscapes,

text: averaging

text: them

text: and

text: calculating

text: distances

text: between

text: (average)

text: persistence

text: landscapes.

text: We

text: have

text: implemented

text: these

text: algorithms

text: and

text: demonstrated

text: how

text: they

text: may

text: be

text: used

text: for

text: hypothesis

text: testing

text: and

text: classiﬁcation.

text: We

text: hope

text: that

text: they

text: will

text: of

text: use

text: to

text: the

text: (topological)

text: data

text: scientist.

text: Furthermore,

text: this

text: is

text: not

text: intended

text: to

text: be

text: the

text: end

text: of

text: the

text: story.

text: In

text: future

text: work,

text: we

text: aim

text: to

text: improve

text: these

text: algorithms

text: and

text: related

text: software

text: and

text: increase

text: their

text: usefulness.

text: Appendix

text: A.

text: Persistence

text: landscapes

text: for

text: barcodes

text: with

text: infinite

text: intervals

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 PETER BUBENIK AND PAWEL(cid:32) DL(cid:32) OTKO Algorithm 2: Compute the persistence landscape using a grid. { } Input: (b , d ) n – a list of pairs b < d which are elements of 0, 2, 4, . . . , 2m; i i i=1 i i × Output: V – the persistence landscape, a two-dimensional array of size K (2m + 1); Initialize W, an array of size 2m + 1 of empty lists of integers; for i = 1 to n do − for j = 1 to di bi do 2 Append j to W[b + j]; i end − − for j = 1 to di bi 1 do 2 − Append j to W[d j]; i end end for i = 0 to 2m do Sort W[i] in decreasing order; end ← K max ∈{ } length of W[i]; i 0,...,2m × Initialize V as a K (2m + 1) zero matrix; for i = 0 to 2m do for k = 0 to length of W[i] do V[k][i] = W[k][i]; end end number can be quadratic in n in the worst case as shown in the Figure 2. This analysis shows that (cid:0) (cid:1) the algorithms presented in this section are optimal in the worst case. In fact, it is easy to see that if n is the number of intervals given by the birth-death pairs, and p is the number of pairwise ≤ ≤ non-empty intersections of these intervals, then P = 3n + 2p. Since p n , we have P n2 + 2n. 2 Figure 2. The worst case scenario for the complexity of the persistence land- scapes occurs when all the pairs of persistence intervals have nonempty intersec- tion. Suppose n persistence intervals having nonempty intersection are given. Then there are n nonzero persistence landscapes and the landscape λ has − k 2n + 3 2k critical points. That gives n2 + 2n = O(n2) critical points in to- tal. In this section, we present algorithms for calculating linear combinations of persistence land- scapes. { R → R} {L } In Section we encoded a persistence landscape λ = λ : by lists of pairs of X Y k k extended real numbers. Fix k. Deﬁne and to be the vectors of critical numbers and critical k k L Y X values obtained from the ﬁrst and second coordinates of elements of . Then = λ ( ), and X Y k k k k λ can be obtained from and by linear interpolation. k k k ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: PETER

text: BUBENIK

text: AND

text: OTKO

text: Algorithm

text: Compute

text: the

text: persistence

text: landscape

text: using

text: a

text: grid.

text: Input:

text: (b

text: d

text: n

text: a

text: list

text: of

text: pairs

text: b

text: d

text: which

text: are

text: elements

text: of

text: i

text: i

text: i

text: i

text: Output:

text: V

text: the

text: persistence

text: landscape,

text: a

text: two-dimensional

text: array

text: of

text: size

text: K

text: Initialize

text: W,

text: an

text: array

text: of

text: size

text: of

text: empty

text: lists

text: of

text: integers;

text: for

text: i

text: to

text: n

text: do

text: for

text: j

text: to

text: di

text: bi

text: do

text: Append

text: j

text: to

text: W[b

text: j];

text: i

text: end

text: for

text: j

text: to

text: di

text: bi

text: do

text: Append

text: j

text: to

text: W[d

text: j];

text: i

text: end

text: end

text: for

text: i

text: to

text: do

text: Sort

text: W[i]

text: in

text: decreasing

text: order;

text: end

text: K

text: max

text: length

text: of

text: W[i];

text: i

text: Initialize

text: V

text: as

text: a

text: K

text: zero

text: matrix;

text: for

text: i

text: to

text: do

text: for

text: k

text: to

text: length

text: of

text: W[i]

text: do

text: V[k][i]

text: W[k][i];

text: end

text: end

text: number

text: can

text: be

text: quadratic

text: in

text: n

text: in

text: the

text: worst

text: case

text: as

text: shown

text: in

text: the

text: Figure

text: This

text: analysis

text: shows

text: that

text: the

text: algorithms

text: presented

text: in

text: this

text: section

text: are

text: optimal

text: in

text: the

text: worst

text: case.

text: In

text: fact,

text: it

text: is

text: easy

text: to

text: see

text: that

text: if

text: n

text: is

text: the

text: number

text: of

text: intervals

text: given

text: by

text: the

text: birth-death

text: pairs,

text: and

text: p

text: is

text: the

text: number

text: of

text: pairwise

text: non-empty

text: intersections

text: of

text: these

text: intervals,

text: then

text: P

text: Since

text: p

text: n

text: we

text: have

text: P

text: Figure

text: The

text: worst

text: case

text: scenario

text: for

text: the

text: complexity

text: of

text: the

text: persistence

text: land-

text: scapes

text: occurs

text: when

text: all

text: the

text: pairs

text: of

text: persistence

text: intervals

text: have

text: nonempty

text: intersec-

text: tion.

text: Suppose

text: n

text: persistence

text: intervals

text: having

text: nonempty

text: intersection

text: are

text: given.

text: Then

text: there

text: are

text: n

text: nonzero

text: persistence

text: landscapes

text: and

text: the

text: landscape

text: λ

text: has

text: k

text: critical

text: points.

text: That

text: gives

text: critical

text: points

text: in

text: to-

text: tal.

text: In

text: this

text: section,

text: we

text: present

text: algorithms

text: for

text: calculating

text: linear

text: combinations

text: of

text: persistence

text: land-

text: scapes.

text: R

text: R}

text: {L

text: In

text: Section

text: we

text: encoded

text: a

text: persistence

text: landscape

text: λ

text: λ

text: by

text: lists

text: of

text: pairs

text: of

text: X

text: Y

text: k

text: k

text: extended

text: real

text: numbers.

text: Fix

text: k.

text: Deﬁne

text: and

text: to

text: be

text: the

text: vectors

text: of

text: critical

text: numbers

text: and

text: critical

text: k

text: k

text: L

text: Y

text: X

text: values

text: obtained

text: from

text: the

text: ﬁrst

text: and

text: second

text: coordinates

text: of

text: elements

text: of

text: Then

text: λ

text: and

text: X

text: Y

text: k

text: k

text: k

text: k

text: λ

text: can

text: be

text: obtained

text: from

text: and

text: by

text: linear

text: interpolation.

text: k

text: k

text: k

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ON A THEORY OF THE B-FUNCTION IN POSITIVE CHARACTERISTIC THOMAS BITOUN Abstract. We introduce a theory of the b-function (or Bernstein-Sato polyno- mial) in positive characteristic. Let k be a ﬁeld of characteristic p > 0 and let ∈ f k[x , . . . , x ] be a nonconstant polynomial. The b-function of f is an ideal 1 n of a nonnoetherian commutative algebra of characteristic p, but it has “roots” in Z . We prove the existence of the b-function as well as the rationality of its roots. p The framework of the theory is that of unit F-modules. There is a close con- nection with test ideals. In particular, we prove that the roots of the b-function ∩ Z are the opposites of the F-jumping exponents of f which are in (0, 1] . (p) Contents Introduction 1 1. The construction 4 2. The relation to test ideals 9 3. Examples 15 References 17 Introduction ∈ C Let Q [x , . . . , x ] be a polynomial. The b-function or Bernstein-Sato poly- 1 n D nomial of Q ([2], [14]) is of foundational importance in -module theory. It is related to ζ-functions ([2], [15]), to the nearby and vanishing cycles of Q ([1], [7], [11]), as well as to many invariants of its singularities, see [13] for a survey. The goal of this note is to present a theory of the b-function in positive charac- teristic. Let us ﬁrst recall the classical complex theory, see for example [9, Chapter 6]. ∈ C D Let Q [x , . . . , x ] be a nonconstant polynomial. We denote by the ring D 1 n A D C ⊗ D A of polynomial diﬀerential operators on n and let [s] := [s] C , where s n is a central parameter. Date: 1 ##LTLine##

text: ON

text: A

text: THEORY

text: OF

text: THE

text: B-FUNCTION

text: IN

text: POSITIVE

text: CHARACTERISTIC

text: THOMAS

text: BITOUN

text: Abstract.

text: We

label: abstract

text: introduce

text: a

text: theory

text: of

text: the

text: b-function

text: (or

text: Bernstein-Sato

text: polyno-

text: mial)

text: in

text: positive

text: characteristic.

text: Let

text: k

text: be

text: a

text: ﬁeld

text: of

text: characteristic

text: p

text: and

text: let

text: f

text: k[x

text: x

text: be

text: a

text: nonconstant

text: polynomial.

text: The

text: b-function

text: of

text: f

text: is

text: an

text: ideal

text: n

text: of

text: a

text: nonnoetherian

text: commutative

text: algebra

text: of

text: characteristic

text: p,

text: but

text: it

text: has

text: “roots”

text: in

text: Z

text: We

text: prove

text: the

text: existence

text: of

text: the

text: b-function

text: as

text: well

text: as

text: the

text: rationality

text: of

text: its

text: roots.

text: p

text: The

text: framework

text: of

text: the

text: theory

text: is

text: that

text: of

text: unit

text: F-modules.

text: There

text: is

text: a

text: close

text: con-

text: nection

text: with

text: test

text: ideals.

text: In

text: particular,

text: we

text: prove

text: that

text: the

text: roots

text: of

text: the

text: b-function

text: Z

text: are

text: the

text: opposites

text: of

text: the

text: F-jumping

text: exponents

text: of

text: f

text: which

text: are

text: in

text: (p)

text: Contents

text: Introduction

text: The

text: construction

text: The

text: relation

text: to

text: test

text: ideals

text: Examples

text: References

text: Introduction

text: C

text: Let

text: Q

text: [x

text: x

text: be

text: a

text: polynomial.

text: The

text: b-function

text: or

text: Bernstein-Sato

text: poly-

text: n

text: D

text: nomial

text: of

text: Q

text: is

text: of

text: foundational

text: importance

text: in

text: -module

text: theory.

text: It

text: is

text: related

text: to

text: ζ-functions

text: to

text: the

text: nearby

text: and

text: vanishing

text: cycles

text: of

text: Q

text: as

text: well

text: as

text: to

text: many

text: invariants

text: of

text: its

text: singularities,

text: see

text: for

text: a

text: survey.

text: The

text: goal

text: of

text: this

text: note

text: is

text: to

text: present

text: a

text: theory

text: of

text: the

text: b-function

text: in

text: positive

text: charac-

text: teristic.

text: Let

text: us

text: ﬁrst

text: recall

text: the

text: classical

text: complex

text: theory,

text: see

text: for

text: example

text: Chapter

text: C

text: D

text: Let

text: Q

text: [x

text: x

text: be

text: a

text: nonconstant

text: polynomial.

text: We

text: denote

text: by

text: the

text: ring

text: D

text: n

text: A

text: D

text: C

text: D

text: A

text: of

text: polynomial

text: diﬀerential

text: operators

text: on

text: n

text: and

text: let

text: [s]

text: [s]

text: C

text: where

text: s

text: n

text: is

text: a

text: central

text: parameter.

text: Date:

text: ##LTLine##


text: ON A THEORY OF THE B-FUNCTION IN POSITIVE CHARACTERISTIC 17 References [1] A. A. Be˘ılinson. How to glue perverse sheaves. In K-theory, arithmetic and geometry (Moscow, 1984–1986), volume 1289 of Lecture Notes in Math., pages 42–51. Springer, Berlin, 1987. [2] I. N. Bernˇste˘ın. Analytic continuation of generalized functions with respect to a parameter. Funkcional. Anal. i Priloˇzen., 6(4):26–40, 1972. [3] Manuel Blickle, Mircea Musta¸tˇa, and Karen E. Smith. Discreteness and rationality of F- thresholds. Michigan Math. J., 57:43–61, 2008. Special volume in honor of Melvin Hochster. [4] Lawrence Ein, Robert Lazarsfeld, Karen E. Smith, and Dror Varolin. Jumping coeﬃcients of multiplier ideals. Duke Math. J., 123(3):469–506, 2004. [5] Matthew Emerton and Mark Kisin. The Riemann-Hilbert correspondence for unit F- crystals. Ast´erisque, (293):vi+257, 2004. [6] Nobuo Hara and Ken-Ichi Yoshida. A generalization of tight closure and multiplier ideals. Trans. Amer. Math. Soc., 355(8):3143–3174 (electronic), 2003. [7] M. Kashiwara. Vanishing cycle sheaves and holonomic systems of diﬀerential equations. In Algebraic geometry (Tokyo/Kyoto, 1982), volume 1016 of Lecture Notes in Math., pages 134–142. Springer, Berlin, 1983. [8] Masaki Kashiwara. B-functions and holonomic systems. Rationality of roots of B-functions. Invent. Math., 38(1):33–53, 1976/77. [9] Masaki Kashiwara. D-modules and microlocal calculus, volume 217 of Translations of Mathe- matical Monographs. American Mathematical Society, Providence, RI, 2003. Translated from the 2000 Japanese original by Mutsumi Saito, Iwanami Series in Modern Mathematics. [10] Gennady Lyubeznik. F-modules: applications to local cohomology and D-modules in char- acteristic p > 0. J. Reine Angew. Math., 491:65–130, 1997. [11] B. Malgrange. Polynˆomes de Bernstein-Sato et cohomologie ´evanescente. In Analysis and topology on singular spaces, II, III (Luminy, 1981), volume 101 of Ast´erisque, pages 243–267. Soc. Math. France, Paris, 1983. [12] Mircea Musta¸t˘a. Bernstein-Sato polynomials in positive characteristic. J. Algebra, 321(1):128–151, 2009. [13] Morihiko Saito. Introduction to a theory of b-functions. arXiv preprint arXiv:math/0610783, 2006. [14] M. Sato. Theory of prehomogeneous vector spaces (notes by T. Shintani). Sugaku no Ayumi, 15(1), 1970. [15] Mikio Sato and Takuro Shintani. On zeta functions associated with prehomogeneous vector spaces. Ann. of Math. (2), 100:131–170, 1974. Higher School of Economics, Moscow Email address: tbitoun@gmail.com

text: ON

text: A

text: THEORY

text: OF

text: THE

text: B-FUNCTION

text: IN

text: POSITIVE

text: CHARACTERISTIC

text: References

label: reference

text: A.

text: A.

text: Be˘ılinson.

text: How

text: to

text: glue

text: perverse

text: sheaves.

text: In

text: K-theory,

text: arithmetic

text: and

text: geometry

text: (Moscow,

text: volume

text: of

text: Lecture

text: Notes

text: in

text: Math.,

text: pages

text: Springer,

text: Berlin,

text: I.

text: N.

text: Bernˇste˘ın.

text: Analytic

text: continuation

text: of

text: generalized

text: functions

text: with

text: respect

text: to

text: a

text: parameter.

text: Funkcional.

text: Anal.

text: i

text: Priloˇzen.,

text: Manuel

text: Blickle,

text: Mircea

text: Musta¸tˇa,

text: and

text: Karen

text: E.

text: Smith.

text: Discreteness

text: and

text: rationality

text: of

text: F-

text: thresholds.

text: Michigan

text: Math.

text: J.,

text: Special

text: volume

text: in

text: honor

text: of

text: Melvin

text: Hochster.

text: Lawrence

text: Ein,

text: Robert

text: Lazarsfeld,

text: Karen

text: E.

text: Smith,

text: and

text: Dror

text: Varolin.

text: Jumping

text: coeﬃcients

text: of

text: multiplier

text: ideals.

text: Duke

text: Math.

text: J.,

text: Matthew

text: Emerton

text: and

text: Mark

text: Kisin.

text: The

text: Riemann-Hilbert

text: correspondence

text: for

text: unit

text: F-

text: crystals.

text: Ast´erisque,

text: Nobuo

text: Hara

text: and

text: Ken-Ichi

text: Yoshida.

text: A

text: generalization

text: of

text: tight

text: closure

text: and

text: multiplier

text: ideals.

text: Trans.

text: Amer.

text: Math.

text: Soc.,

text: (electronic),

text: M.

text: Kashiwara.

text: Vanishing

text: cycle

text: sheaves

text: and

text: holonomic

text: systems

text: of

text: diﬀerential

text: equations.

text: In

text: Algebraic

text: geometry

text: (Tokyo/Kyoto,

text: volume

text: of

text: Lecture

text: Notes

text: in

text: Math.,

text: pages

text: Springer,

text: Berlin,

text: Masaki

text: Kashiwara.

text: B-functions

text: and

text: holonomic

text: systems.

text: Rationality

text: of

text: roots

text: of

text: B-functions.

text: Invent.

text: Math.,

text: Masaki

text: Kashiwara.

text: D-modules

text: and

text: microlocal

text: calculus,

text: volume

text: of

text: Translations

text: of

text: Mathe-

text: matical

text: Monographs.

text: American

text: Mathematical

text: Society,

text: Providence,

text: RI,

text: Translated

text: from

text: the

text: Japanese

text: original

text: by

text: Mutsumi

text: Saito,

text: Iwanami

text: Series

text: in

text: Modern

text: Mathematics.

text: Gennady

text: Lyubeznik.

text: F-modules:

text: applications

text: to

text: local

text: cohomology

text: and

text: D-modules

text: in

text: char-

text: acteristic

text: p

text: J.

text: Reine

text: Angew.

text: Math.,

text: B.

text: Malgrange.

text: Polynˆomes

text: de

text: Bernstein-Sato

text: et

text: cohomologie

text: ´evanescente.

text: In

text: Analysis

text: and

text: topology

text: on

text: singular

text: spaces,

text: II,

text: III

text: (Luminy,

text: volume

text: of

text: Ast´erisque,

text: pages

text: Soc.

text: Math.

text: France,

text: Paris,

text: Mircea

text: Musta¸t˘a.

text: Bernstein-Sato

text: polynomials

text: in

text: positive

text: characteristic.

text: J.

text: Algebra,

text: Morihiko

text: Saito.

text: Introduction

text: to

text: a

text: theory

text: of

text: b-functions.

text: arXiv

text: preprint

text: M.

text: Sato.

text: Theory

text: of

text: prehomogeneous

text: vector

text: spaces

text: (notes

text: by

text: T.

text: Shintani).

text: Sugaku

text: no

text: Ayumi,

text: Mikio

text: Sato

text: and

text: Takuro

text: Shintani.

text: On

text: zeta

text: functions

text: associated

text: with

text: prehomogeneous

text: vector

text: spaces.

text: Ann.

text: of

text: Math.

text: Higher

text: School

text: of

text: Economics,

text: Moscow

text: Email

text: address:

text: tbitoun@gmail.com


text: ON A THEORY OF THE B-FUNCTION IN POSITIVE CHARACTERISTIC 7 ∪ Note that Mf = l≥0Mf(l) and Nf = l−i→ml≥0 Nf(l), where the limit is taken over the above morphisms. O Let s be a global section of . We denote by D(l)s the left D(l)-submodule of O X X X generated by s. X By explicit computations, Musta¸t˘a observed the following: ∈ N (cid:77) Proposition 1. Let l . Then there is a natural isomorphism of left D(l)- X modules, ∼ M(l) = D(l)fn. f X ≤ (cid:77)0 n<pl+1 It induces an isomorphism: ∼ N(l) = D(l)fn/D(l)fn+1. f X X ≤ (cid:80) 0 n<pl+1 Hence the higher Euler operators act on the right-hand side by transport of struc- ture. The actions are as follow: If the base p expansion of n is n = ≤ a pe, ∀ − 0 e<l+1 e then e < l + 1, νe acts o(cid:77)n DX(l)fn/DX(l)⊂fn+1(cid:77)by ae. Finally, the natural inclusions M(l) M(l+1) transport to the injections: f f → D(l)fn (cid:18) (cid:19)D(l+1)fm (cid:88)X X ≤ ≤ 0 n<pl+1 0 m<pl+2 − (cid:55)→ − p 1 g g. ( 1)j fjpl+1. j ≤ 0 j<p Proof. The ﬁrst part of the statement is [12, Proposition 6.1.], the second is [12, (cid:3) Corollary 6.5.] and the last is [12, Remark 5.7.]. We would like to express the N(l) as Frobenius pullbacks of a coherent sheaf. f This coherent sheaf is expressed in terms of the test ideals of f. ∈ R ⊂ O Let λ ≥ and let τ(fλ) be the test ideal of exponent λ of f. The test 0 X ⊂ O ideals form a decreasing sequence of ideals . A F -jumping exponent of f is a ∈ R ∀ X − (cid:54) positive real number λ ≥ , such that, (cid:15) > 0, τ(fλ (cid:15)) = τ(fλ), see [12, Section 0 3] for deﬁnitions. They satisfy the following ﬁniteness theorem: Theorem 1. ([3]) The set of F -jumping exponents of f is R (1) a discrete subset of Q (2) a subset of .

text: ON

text: A

text: THEORY

text: OF

text: THE

text: B-FUNCTION

text: IN

text: POSITIVE

text: CHARACTERISTIC

text: Note

text: that

text: Mf

text: and

text: Nf

text: Nf(l),

text: where

text: the

text: limit

text: is

text: taken

text: over

text: the

text: above

text: morphisms.

text: O

text: Let

text: s

text: be

text: a

text: global

text: section

text: of

text: We

text: denote

text: by

text: D(l)s

text: the

text: left

text: D(l)-submodule

text: of

text: O

text: X

text: X

text: X

text: generated

text: by

text: s.

text: X

text: By

text: explicit

text: computations,

text: Musta¸t˘a

text: observed

text: the

text: following:

text: N

text: Proposition

text: Let

text: l

text: Then

text: there

text: is

text: a

text: natural

text: isomorphism

text: of

text: left

text: D(l)-

text: X

text: modules,

text: M(l)

text: D(l)fn.

text: f

text: X

text: It

text: induces

text: an

text: isomorphism:

text: N(l)

text: f

text: X

text: X

text: Hence

text: the

text: higher

text: Euler

text: operators

text: act

text: on

text: the

text: right-hand

text: side

text: by

text: transport

text: of

text: struc-

text: ture.

text: The

text: actions

text: are

text: as

text: follow:

text: If

text: the

text: base

text: p

text: expansion

text: of

text: n

text: is

text: n

text: a

text: pe,

text: e

text: then

text: e

text: l

text: νe

text: acts

text: ae.

text: Finally,

text: the

text: natural

text: inclusions

text: M(l)

text: transport

text: to

text: the

text: injections:

text: f

text: f

text: D(l)fn

text: X

text: p

text: g

text: g.

text: j

text: j<p

text: Proof.

text: The

text: ﬁrst

text: part

text: of

text: the

text: statement

text: is

text: Proposition

text: the

text: second

text: is

text: Corollary

text: and

text: the

text: last

text: is

text: Remark

text: We

text: would

text: like

text: to

text: express

text: the

text: N(l)

text: as

text: Frobenius

text: pullbacks

text: of

text: a

text: coherent

text: sheaf.

text: f

text: This

text: coherent

text: sheaf

text: is

text: expressed

text: in

text: terms

text: of

text: the

text: test

text: ideals

text: of

text: f.

text: R

text: O

text: Let

text: λ

text: and

text: let

text: τ(fλ)

text: be

text: the

text: test

text: ideal

text: of

text: exponent

text: λ

text: of

text: f.

text: The

text: test

text: X

text: O

text: ideals

text: form

text: a

text: decreasing

text: sequence

text: of

text: ideals

text: A

text: F

text: -jumping

text: exponent

text: of

text: f

text: is

text: a

text: R

text: X

text: positive

text: real

text: number

text: λ

text: such

text: that,

text: τ(fλ

text: τ(fλ),

text: see

text: Section

text: for

text: deﬁnitions.

text: They

text: satisfy

text: the

text: following

text: ﬁniteness

text: theorem:

text: Theorem

text: The

text: set

text: of

text: F

text: -jumping

text: exponents

text: of

text: f

text: is

text: R

text: a

text: discrete

text: subset

text: of

text: Q

text: a

text: subset

text: of


text: and as t → −∞ we have that κ → ∞ a.s. and(cid:95)B∅ ↑ Aˆ , it follows by monotone convergence that 0 ,r r ∞ (cid:44) →lim−∞ W∅(n,t0) = Uˆr W (n) a.s. t0 r=0 ∞ Therefore, it only remains to verify that W (n) < a.s. To establish the ﬁniteness of W (n) we note that it suﬃces to show that P (Uˆ > 0 i.o.) = 0. r This in turn will follow from the Borel-Cantelli Lemma once we show that ≤ P (Uˆ > 0) cr (6.2) r for some constant 0 < c < 1. Therefore, we focus on showing (6.2). { } Recall from the observations m(cid:0) ade(cid:1) at the beginning of this section that the τˆ ∈G are neither { } i i n(t0) i.i.d. nor independent of the Nˆ ∈G . More precisely, recall that for each piece of a job requiring i−i n(t0) service at server s there are n−1 k! possible immediate predecessors of size k (i.e., jobs that also i k 1 require service from server s ). Since the arrival of jobs is assumed to follow a Poisson process, i this leads in (3.4) to the inter arrival time between a ﬁxed piece of a job and its unique immediate ∗ predecessor to be exponentially distributed with rate λ . The problem arises when the piece of a n job has two or more immediate predecessors, in which case the rate for the exponential changes. (cid:0) (cid:1) Consider an arrival that is predecessor to two jobs (or two pieces of the same job), j and j , and 1 2 note that there must be two diﬀerent servers, say s and s , that are required by the arriving job i1 i2 − and that are also assigned to jobs j and j , respectively. There are only n−2 k! possible jobs of 1 2 (cid:18) (cid:19) k 2 size k requiring service by servers s and s(cid:88), and therefore, the rate at which such a predecessor i1 i2 arrives is given by − mn n 2 λ(2) = λ − k!. n k k 2 (cid:18) k(cid:19)=2 (cid:18) (cid:19) (cid:88) (cid:88) In general, a job that is predecessor to jobs j , j , . . . , j in the graph arrives at a rate 1 2 r − − mn n r ≤ mn n 1 ∗ λ(r) = λ − k! λ − k! = λ . n k k r k k 1 n k=r k=1 { } As for the lack of independence between the τˆ , note that the inter arrival times between pieces i { } of jobs that have a common immediate predecessor are dependent. The sequence τˆ is also { } i dependent on the Nˆ , since a large number of jobs awaiting for a predecessor to arrive increases i the probability of an arriving job being predecessor to two or more pieces at a time. Hence, the analysis of P (Uˆ > 0) needs some care. r (cid:32) (cid:33)     (cid:16) (cid:17) (cid:95) (cid:88) We start by using Markov’s inequality to obtain     ≤ ≤ P Uˆ > 0 = P max Sˆ > 0 E eβSˆj E eβSˆj . r ∈ j j Aˆr j∈Aˆr j∈Aˆr 22

text: and

text: as

text: t

text: we

text: have

text: that

text: κ

text: a.s.

text: Aˆ

text: it

text: follows

text: by

text: monotone

text: convergence

text: that

text: ,r

text: r

text: →lim−∞

text: Uˆr

text: W

text: (n)

text: a.s.

text: Therefore,

text: it

text: only

text: remains

text: to

text: verify

text: that

text: W

text: (n)

text: a.s.

text: To

text: establish

text: the

text: ﬁniteness

text: of

text: W

text: (n)

text: we

text: note

text: that

text: it

text: suﬃces

text: to

text: show

text: that

text: P

text: (Uˆ

text: i.o.)

text: r

text: This

text: in

text: turn

text: will

text: follow

text: from

text: the

text: Borel-Cantelli

text: Lemma

text: once

text: we

text: show

text: that

label: equation

text: P

text: (Uˆ

text: cr

text: r

text: for

text: some

text: constant

text: c

text: Therefore,

text: we

text: focus

text: on

text: showing

text: Recall

text: from

text: the

text: observations

text: at

text: the

text: beginning

text: of

text: this

text: section

text: that

text: the

text: τˆ

text: ∈G

text: are

text: neither

text: i

text: i

text: i.i.d.

text: nor

text: independent

text: of

text: the

text: Nˆ

text: ∈G

text: More

text: precisely,

text: recall

text: that

text: for

text: each

text: piece

text: of

text: a

text: job

text: requiring

text: i−i

text: service

text: at

text: server

text: s

text: there

text: are

text: k!

text: possible

text: immediate

text: predecessors

text: of

text: size

text: k

text: (i.e.,

text: jobs

text: that

text: also

text: i

text: k

text: require

text: service

text: from

text: server

text: s

text: Since

text: the

text: arrival

text: of

text: jobs

text: is

text: assumed

text: to

text: follow

text: a

text: Poisson

text: process,

text: i

text: this

text: leads

text: in

text: to

text: the

text: inter

text: arrival

text: time

text: between

text: a

text: ﬁxed

text: piece

text: of

text: a

text: job

text: and

text: its

text: unique

text: immediate

text: predecessor

text: to

text: be

text: exponentially

text: distributed

text: with

text: rate

text: λ

text: The

text: problem

text: arises

text: when

text: the

text: piece

text: of

text: a

text: n

text: job

text: has

text: two

text: or

text: more

text: immediate

text: predecessors,

text: in

text: which

text: case

text: the

text: rate

text: for

text: the

text: exponential

text: changes.

text: Consider

text: an

text: arrival

text: that

text: is

text: predecessor

text: to

text: two

text: jobs

text: (or

text: two

text: pieces

text: of

text: the

text: same

text: job),

text: j

text: and

text: j

text: and

text: note

text: that

text: there

text: must

text: be

text: two

text: diﬀerent

text: servers,

text: say

text: s

text: and

text: s

text: that

text: are

text: required

text: by

text: the

text: arriving

text: job

text: and

text: that

text: are

text: also

text: assigned

text: to

text: jobs

text: j

text: and

text: j

text: respectively.

text: There

text: are

text: only

text: k!

text: possible

text: jobs

text: of

text: k

text: size

text: k

text: requiring

text: service

text: by

text: servers

text: s

text: and

text: and

text: therefore,

text: the

text: rate

text: at

text: which

text: such

text: a

text: predecessor

text: arrives

text: is

text: given

text: by

text: mn

text: n

text: λ

text: k!.

text: n

text: k

text: k

text: In

text: general,

text: a

text: job

text: that

text: is

text: predecessor

text: to

text: jobs

text: j

text: j

text: j

text: in

text: the

text: graph

text: arrives

text: at

text: a

text: rate

text: r

text: mn

text: n

text: r

text: mn

text: n

text: λ(r)

text: λ

text: k!

text: λ

text: k!

text: λ

text: n

text: k

text: k

text: r

text: k

text: k

text: n

text: k=r

text: As

text: for

text: the

text: lack

text: of

text: independence

text: between

text: the

text: τˆ

text: note

text: that

text: the

text: inter

text: arrival

text: times

text: between

text: pieces

text: i

text: of

text: jobs

text: that

text: have

text: a

text: common

text: immediate

text: predecessor

text: are

text: dependent.

text: The

text: sequence

text: τˆ

text: is

text: also

text: i

text: dependent

text: on

text: the

text: Nˆ

text: since

text: a

text: large

text: number

text: of

text: jobs

text: awaiting

text: for

text: a

text: predecessor

text: to

text: arrive

text: increases

text: i

text: the

text: probability

text: of

text: an

text: arriving

text: job

text: being

text: predecessor

text: to

text: two

text: or

text: more

text: pieces

text: at

text: a

text: time.

text: Hence,

text: the

text: analysis

text: of

text: P

text: (Uˆ

text: needs

text: some

text: care.

text: r

text: We

text: start

text: by

text: using

text: Markov’s

text: inequality

text: to

text: obtain

text: P

text: Uˆ

text: P

text: max

text: Sˆ

text: E

text: eβSˆj

text: E

text: eβSˆj

text: r

text: j

text: j

text: Aˆr

text: j∈Aˆr

text: j∈Aˆr


text: { } Now note that along a path all jobs are diﬀerent and therefore the service requirements χ | r are { } − i k{ k=1} i.i.d. with distribution B, and are independent of the Nˆ | r 1. The inter arrival times τˆ| r { } − i k k=0 i k k=1 do depend on the Nˆ | r 1 in the sense that a large number of jobs in the predecessor graph at i k k=0 the time a job arrives increases its probability of being an immediate predecessor to more than one job, and therefore inﬂuences(cid:104) the rate of th(cid:105)e cor(cid:16)res(cid:104)ponding(cid:105)τˆ(cid:17). It follows that if we replace them by i.i.d. copies of τ˜ independent of everything else, we obtain E eβSˆi1(i ∈ M ) ≤ E eβ(χ−τ˜) r P (i ∈ M ). r r C To compute the last probability let denote the event that i is a common immediate predecessor i (cid:88) (cid:88) to two or more jobs/fragments in the predecessor graph, and note that r r ∈ | − ∈ | ∈ ≤ ∈ C 1(i M ) = 1((i s 1) Mc− , (i s) M ) 1(i A , | ), r s 1 s r i s s=1 s=1 (cid:34) (cid:35) (cid:88) (cid:88) (cid:88) (cid:89) and therefore, − r r 1 ∈ ≤ ≤ C P (i M ) E 1(i Nˆ | )1( | ) . r k+1 i k i s ∈N ∈N i r s=1 i r k=0 + + − − · · · − Next, let σ = T τˆ| τˆ denote the time at which job i arrived to the predecessor graph. F i 1 i 1 i Deﬁne = σ((Nˆ , s , χ , τˆ ) : σ > t) to be the sigma algebra containing the “history” of the t j j j j j F C pnroerdtehceesvsaolruegroafpNh(cid:88)ˆ o.vNero(cid:34)wth(cid:89)enointetetrhvaatl f(ot,rT11]≤, asn≤d nr(cid:35)o−te1,that σi does not reveal whether i occurred i − r 1 ≤ C E (cid:34) 1(i Nˆ | )1( | (cid:34)) (cid:12) (cid:35)(cid:35) (cid:88) (cid:89) k+1 i k i s (cid:89) (cid:12) ∈N (cid:12) i r k=0 (cid:12) + − − (cid:12) s 1 r 1 ≤ ≤ C F = ∈N E 1(ik+1 Nˆi|k)E 1(ik+1 Nˆi|k)1( i|s) σi|s . i r k=0 k=s + (cid:34) { } − (cid:12) (cid:35) F Moreove(cid:89)r, since Nˆi|k rk=1s are in(cid:12)(cid:12)(cid:12)dependent o(cid:104)f σi|s, then (cid:12)(cid:12)(cid:12) (cid:105) (cid:89) − (cid:12) − r 1 r 1 ≤ C F ≤ C F ≤ E 1(ik+1 Nˆi|k)(cid:81)1( i|s) σi|s = E 1(is+1 Nˆi|s)1( i|s) σi|s P (ik+1 Nˆi|k), k=s k=s+1 ≡ with the convention that b x 1 if a > b. i=a i To analyze the last conditional expectation let K be the number of pieces of jobs that are available t at time t, where by available we mean that they they do not have an immediate predecessor in C ∈ N (t, T ]. Note that the event is a function of K(cid:0) and(cid:1)(cid:0)Nˆ o(cid:1)nly; more precisely, for any j s we 1 j σj j + have (cid:0) (cid:1)(cid:0) (cid:1) − − P (Cj|Nˆj, Kσj) = KKσ1jσj 1 nN−nˆ−j1−22 ≤ Nnˆj . 1 Nˆj 1 26 ##LTLine## ##LTLine##

text: Now

text: note

text: that

text: along

text: a

text: path

text: all

text: jobs

text: are

text: diﬀerent

text: and

text: therefore

text: the

text: service

text: requirements

text: χ

text: r

text: are

text: i

text: k{

text: i.i.d.

text: with

text: distribution

text: B,

text: and

text: are

text: independent

text: of

text: the

text: Nˆ

text: r

text: The

text: inter

text: arrival

text: times

text: τˆ|

text: r

text: i

text: k

text: i

text: k

text: do

text: depend

text: on

text: the

text: Nˆ

text: r

text: in

text: the

text: sense

text: that

text: a

text: large

text: number

text: of

text: jobs

text: in

text: the

text: predecessor

text: graph

text: at

text: i

text: k

text: the

text: time

text: a

text: job

text: arrives

text: increases

text: its

text: probability

text: of

text: being

text: an

text: immediate

text: predecessor

text: to

text: more

text: than

text: one

text: job,

text: and

text: therefore

text: the

text: rate

text: of

text: It

text: follows

text: that

text: if

text: we

text: replace

text: them

text: by

text: i.i.d.

text: copies

text: of

text: τ˜

text: independent

text: of

text: everything

text: else,

text: we

text: obtain

text: E

text: M

text: E

text: eβ(χ−τ˜)

text: r

text: P

text: (i

text: M

text: r

text: r

text: C

text: To

text: compute

text: the

text: last

text: probability

text: let

text: denote

text: the

text: event

text: that

text: i

text: is

text: a

text: common

text: immediate

text: predecessor

text: i

text: to

text: two

text: or

text: more

text: jobs/fragments

text: in

text: the

text: predecessor

text: graph,

text: and

text: note

text: that

text: r

text: r

text: C

text: M

text: s

text: Mc−

text: (i

text: s)

text: M

text: A

text: r

text: s

text: s

text: r

text: i

text: s

text: and

text: therefore,

text: r

text: r

text: C

text: P

text: (i

text: M

text: E

text: Nˆ

text: r

text: i

text: k

text: i

text: s

text: ∈N

text: ∈N

text: i

text: r

text: i

text: r

text: Next,

text: let

text: σ

text: T

text: τˆ|

text: τˆ

text: denote

text: the

text: time

text: at

text: which

text: job

text: i

text: arrived

text: to

text: the

text: predecessor

text: graph.

text: F

text: i

text: i

text: i

text: Deﬁne

text: σ((Nˆ

text: s

text: χ

text: τˆ

text: σ

text: t)

text: to

text: be

text: the

text: sigma

text: algebra

text: containing

text: the

text: “history”

text: of

text: the

text: t

text: j

text: j

text: j

text: j

text: j

text: F

text: C

text: asn≤d

text: σi

text: does

text: not

text: reveal

text: whether

text: i

text: occurred

text: i

text: r

text: C

text: E

text: Nˆ

text: i

text: k

text: i

text: s

text: ∈N

text: i

text: r

text: s

text: r

text: C

text: F

text: ∈N

text: E

text: Nˆi|k)E

text: i|s)

text: σi|s

text: i

text: r

text: k=s

text: F

text: since

text: Nˆi|k

text: are

text: σi|s,

text: then

text: r

text: r

text: C

text: F

text: C

text: F

text: E

text: i|s)

text: σi|s

text: E

text: i|s)

text: σi|s

text: P

text: Nˆi|k),

text: k=s

text: with

text: the

text: convention

text: that

text: b

text: x

text: if

text: a

text: b.

text: i=a

text: i

text: To

text: analyze

text: the

text: last

text: conditional

text: expectation

text: let

text: K

text: be

text: the

text: number

text: of

text: pieces

text: of

text: jobs

text: that

text: are

text: available

text: t

text: at

text: time

text: t,

text: where

text: by

text: available

text: we

text: mean

text: that

text: they

text: they

text: do

text: not

text: have

text: an

text: immediate

text: predecessor

text: in

text: C

text: N

text: (t,

text: T

text: Note

text: that

text: the

text: event

text: is

text: a

text: function

text: of

text: more

text: precisely,

text: for

text: any

text: j

text: s

text: we

text: j

text: σj

text: j

text: have

text: P

text: (Cj|Nˆj,

text: Kσj)

text: Nnˆj

text: Nˆj

text: ##LTLine##

text: ##LTLine##


text: (cid:104) (cid:12) (cid:105) (cid:104) (cid:12) (cid:105) (cid:12) (cid:12) It follows that (cid:12) (cid:12) (cid:34) (cid:12) (cid:35) ≤ C F ≤ C(cid:12) | F E 1(j Nˆ )1( ) = E 1(j Nˆ )P ( (cid:12) Nˆ , K ) s+1 j j σj s+1 j (cid:12)j j σj σj (cid:12) ≤ ≤ Nˆ F 1 ≤ E 1(j Nˆ ) j = E[Nˆ 1(j Nˆ )]. s+1 j n σj n j s+1 j (cid:34) (cid:35) ≤ ≤ − (cid:88) (cid:89) It follows that, for 1 s r 1, − r 1 ≤ C E (cid:34) 1(i Nˆ | )1( (cid:35)| ) (cid:88) (cid:89) k+1 i k i s (cid:89) ∈N i r k=0 + − − s 1 r 1 ≤ ≤ 1 ≤ ≤ (cid:16) E (cid:17) 1(i Nˆ | ) E[Nˆ | 1(i Nˆ | )] P (i Nˆ | ) ∈N k+1 i k n i s s+1 i s k+1 i k i r k=0 k=s+1 + − 1 r 1 = E[N˜] E[N˜2]. n (cid:34) (cid:35) (cid:88) (cid:89) (cid:88) (cid:89) For s = r note that the same arguments used above give − − r 1 r 1 ≤ C ≤ C E 1(i Nˆ | )1( ) = P (i Nˆ | )P ( ) k+1 i k i (cid:88) (cid:89) k+1 i k i (cid:16) (cid:17) ∈N ∈N i r k=0 i r k=0 + + − ≤ r 1 ≤ E[Nˆ ] 1 r+1 P (i Nˆ | ) i = E[N˜] . ∈N k+1 i k n n i r k=0   + (cid:32) (cid:33) (cid:88) (cid:16) (cid:104) (cid:105)(cid:17) (cid:88) (cid:16) (cid:17) (cid:16) (cid:17) We conclude that   − r 1 − E eβSˆi ≤ E eβ((cid:32)χ−τ˜) r 1 E[N˜] (cid:33)r 1 E[N˜2] + 1 E[N˜] r+1 ∈ n n i Mr s=1 1 − E[N˜2] = (ρ˜ )r (r 1) + E[N˜] . n β E[N˜] ≤ ≤ Noting that 1 (E[N˜])2 E[N˜2] completes the proof. ≥ Lemma 6.3 Let X , X , Y , Y be nonnegative random variables and let p 1. Then 1 2 1 2 | ∨ − ∨ | ≤ | − | (E [ X X Y Y p])1/p (E [Xp])1/p + (E [Y p])1/p + (E [ X Y p])1/p . 1 2 1 2 2 2 1 1 Proof. First note that for any real numbers x , x we have that 1 2 ∨ − x x = (x x )+ + x , 1 2 2 1 1 from where we obtain that for y , y also real, 1 2 | ∨ − ∨ | ≤ − | − | − x x y y (x x )+ + x y + (y y )+. 1 2 1 2 2 1 1 1 2 1 27 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: It

text: follows

text: that

text: C

text: F

text: F

text: E

text: Nˆ

text: E

text: Nˆ

text: )P

text: Nˆ

text: K

text: j

text: j

text: σj

text: j

text: j

text: σj

text: σj

text: Nˆ

text: F

text: E

text: Nˆ

text: j

text: E[Nˆ

text: Nˆ

text: j

text: n

text: σj

text: n

text: j

text: j

text: It

text: follows

text: that,

text: for

text: s

text: r

text: r

text: C

text: E

text: Nˆ

text: i

text: k

text: i

text: s

text: ∈N

text: i

text: r

text: s

text: r

text: E

text: Nˆ

text: E[Nˆ

text: Nˆ

text: P

text: (i

text: Nˆ

text: ∈N

text: i

text: k

text: n

text: i

text: s

text: i

text: s

text: i

text: k

text: i

text: r

text: r

text: E[N˜]

text: n

text: For

text: s

text: r

text: note

text: that

text: the

text: same

text: arguments

text: used

text: above

text: give

text: r

text: r

text: C

text: C

text: E

text: Nˆ

text: P

text: (i

text: Nˆ

text: )P

text: i

text: k

text: i

text: i

text: k

text: i

text: ∈N

text: ∈N

text: i

text: r

text: i

text: r

text: r

text: E[Nˆ

text: P

text: (i

text: Nˆ

text: i

text: E[N˜]

text: ∈N

text: i

text: k

text: n

text: n

text: i

text: r

text: We

text: conclude

text: that

text: r

text: E

text: eβSˆi

text: E

text: r

text: E[N˜]

text: E[N˜]

text: n

text: n

text: i

text: Mr

text: (ρ˜

text: )r

text: (r

text: E[N˜]

text: n

text: β

text: E[N˜]

text: Noting

text: that

text: completes

text: the

text: proof.

text: Lemma

text: Let

text: X

text: X

text: Y

text: Y

text: be

text: nonnegative

text: random

text: variables

text: and

text: let

text: p

text: Then

text: (E

text: X

text: X

text: Y

text: Y

text: (E

text: (E

text: [Y

text: (E

text: X

text: Y

text: Proof.

text: First

text: note

text: that

text: for

text: any

text: real

text: numbers

text: x

text: x

text: we

text: have

text: that

text: x

text: x

text: (x

text: x

text: x

text: from

text: where

text: we

text: obtain

text: that

text: for

text: y

text: y

text: also

text: real,

text: x

text: x

text: y

text: y

text: (x

text: x

text: x

text: y

text: (y

text: y

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ≥ Moreover, provided x , y , x , y 0, we have that 1 1 2 2 | ∨ − ∨ | ≤ | − | x x y y x + x y + y . 1 2 1 2 2 1 1 2 Substituting in the random variables and using Minkowski’s inequality gives the result. The following proposition contains the main coupling between the predecessor graph and its weighted branching tree approximation. Its proof relies on the bound provided by Lemma 6.2. (cid:87) (cid:87) Proposition 6.4 Suppose that Assumption 3.1 is satisﬁed. Then, for νˆ , the probability measure k → ∞ of k Uˆ , and ν˜ , the probability measure of k U˜ , we have that for any r , and any ≥ r=0 r k r=0 r n p 1, → → ∞ W (νˆ , ν˜ ) 0 n . p rn rn Proof. From the deﬁnition of the Wasserstein metric, we need to construct a coupling of νˆ and rn ν˜ for which we can show that their L distance converges to zero. We will do this by deﬁning a rn p weighted branching tree that will be very close to the predecessor graph restricted to predecessors at graph distance at most r of the tagged job (i.e., whose labels are of the form i = (i , . . . , i ) ≤ n { ∈ ∼ ≺ } 1 r with r r ). To start, deﬁne M = i Aˆ : i j for some j i as in Lemma 6.2. To construct n r r the weighed branching tree we proceed inductively starting from the tagged job. { (cid:48)} { (cid:48)} Let N˜ ∈ be a sequence of i.i.d. copies of N˜, let χ ∈ be a sequence of i.i.d. copies of χ, and let { (cid:48)} i i U i i U ∗ τ˜ ∈ be a sequence of i.i.d. exponential random variables with rate λ , all sequences independent i i U n {∅} of e≡ach other and of a(cid:40)ll other{rand≤om ≤varia}bles used up t(cid:40)o now≥. Next, let A˜0∈= = Aˆ0 and N˜∅ Nˆ∅, which deﬁnes A˜ = i : 1 i N˜∅ . In general, for r 1 and each i A˜ , set 1 r ∈ ∈ Nˆ , if i Mc, Xˆ , if i Mc, , N˜i = N˜i(cid:48), otherwisre, and X˜i = χ(cid:48)i− τ˜(cid:48), otherwisre. i i i { } { ∈ ≤ ≤ } Then, use the newly deﬁned N˜ ∈ to construct A˜ = (i, i ) : i A˜ , 1 i N˜ . We i i A⊆˜r ∩ r+1 { r+1 r } r+1 i point out that, by construction, Mc Aˆ A˜ . Also, the (N˜ , X˜ , . . . , X˜ ) ∈ are now i.i.d. r r r i (i,1) (i,N˜i) i U with the same distribution as (N˜, X˜ , . . . , X˜ ), and therefore deﬁne a weighted branching tree. 1 N˜ Recall from the beginning of the section that · · · (cid:54) ∅ S˜∅ = 0, S˜i = X˜i|1 +(cid:95)X˜i|2 + + X˜i, i = , and (cid:104)(cid:12) (cid:12)U˜(cid:105) = S˜ . (cid:12)(cid:87) (cid:87) (cid:12) r i (cid:12) (cid:12) ∈ i A˜r − p → → ∞ We will show that E rn Uˆ rn U˜ 0 as n . r=0 r r=0 r   To this end, note that we can split t(cid:95)he paths(cid:95)in Uˆ and U˜ as(cid:110)follows: (cid:111) r r   (cid:44) Uˆ = max Sˆ , Sˆ+ max Uˆ(1), Uˆ(2) , r i i r r ∈ ∈ i Mrc i Mr 28

text: Moreover,

text: provided

text: x

text: y

text: x

text: y

text: we

text: have

text: that

text: x

text: x

text: y

text: y

text: x

text: x

text: y

text: y

text: Substituting

text: in

text: the

text: random

text: variables

text: and

text: using

text: Minkowski’s

text: inequality

text: gives

text: the

text: result.

text: The

text: following

text: proposition

text: contains

text: the

text: main

text: coupling

text: between

text: the

text: predecessor

text: graph

text: and

text: its

text: weighted

text: branching

text: tree

text: approximation.

text: Its

text: proof

text: relies

text: on

text: the

text: bound

text: provided

text: by

text: Lemma

text: Proposition

text: Suppose

text: that

text: Assumption

text: is

text: satisﬁed.

text: Then,

text: for

text: νˆ

text: the

text: probability

text: measure

text: k

text: of

text: k

text: Uˆ

text: and

text: ν˜

text: the

text: probability

text: measure

text: of

text: k

text: U˜

text: we

text: have

text: that

text: for

text: any

text: r

text: and

text: any

text: r

text: k

text: r

text: n

text: p

text: W

text: (νˆ

text: ν˜

text: n

text: p

text: rn

text: rn

text: Proof.

text: From

text: the

text: deﬁnition

text: of

text: the

text: Wasserstein

text: metric,

text: we

text: need

text: to

text: construct

text: a

text: coupling

text: of

text: νˆ

text: and

text: rn

text: ν˜

text: for

text: which

text: we

text: can

text: show

text: that

text: their

text: L

text: distance

text: converges

text: to

text: zero.

text: We

text: will

text: do

text: this

text: by

text: deﬁning

text: a

text: rn

text: p

text: weighted

text: branching

text: tree

text: that

text: will

text: be

text: very

text: close

text: to

text: the

text: predecessor

text: graph

text: restricted

text: to

text: predecessors

text: at

text: graph

text: distance

text: at

text: most

text: r

text: of

text: the

text: tagged

text: job

text: (i.e.,

text: whose

text: labels

text: are

text: of

text: the

text: form

text: i

text: (i

text: i

text: n

text: r

text: with

text: r

text: r

text: To

text: start,

text: deﬁne

text: M

text: i

text: Aˆ

text: i

text: j

text: for

text: some

text: j

text: i

text: as

text: in

text: Lemma

text: To

text: construct

text: n

text: r

text: r

text: the

text: weighed

text: branching

text: tree

text: we

text: proceed

text: inductively

text: starting

text: from

text: the

text: tagged

text: job.

text: Let

text: N˜

text: be

text: a

text: sequence

text: of

text: i.i.d.

text: copies

text: of

text: N˜,

text: let

text: χ

text: be

text: a

text: sequence

text: of

text: i.i.d.

text: copies

text: of

text: χ,

text: and

text: let

text: i

text: i

text: U

text: i

text: i

text: U

text: τ˜

text: be

text: a

text: sequence

text: of

text: i.i.d.

text: exponential

text: random

text: variables

text: with

text: rate

text: λ

text: all

text: sequences

text: independent

text: i

text: i

text: U

text: n

text: of

text: e≡ach

text: other

text: and

text: of

text: other{rand≤om

text: ≤varia}bles

text: used

text: up

text: now≥.

text: Next,

text: let

text: and

text: N˜∅

text: Nˆ∅,

text: which

text: deﬁnes

text: A˜

text: i

text: i

text: N˜∅

text: In

text: general,

text: for

text: r

text: and

text: each

text: i

text: A˜

text: set

text: r

text: Nˆ

text: if

text: i

text: Mc,

text: Xˆ

text: if

text: i

text: Mc,

text: N˜i

text: otherwisre,

text: and

text: X˜i

text: otherwisre.

text: i

text: i

text: i

text: Then,

text: use

text: the

text: newly

text: deﬁned

text: N˜

text: to

text: construct

text: A˜

text: (i,

text: i

text: i

text: A˜

text: i

text: N˜

text: We

text: i

text: i

text: A⊆˜r

text: r

text: i

text: point

text: out

text: that,

text: by

text: construction,

text: Mc

text: Aˆ

text: A˜

text: Also,

text: the

text: (N˜

text: X˜

text: X˜

text: are

text: now

text: i.i.d.

text: r

text: r

text: r

text: i

text: (i,N˜i)

text: i

text: U

text: with

text: the

text: same

text: distribution

text: as

text: (N˜,

text: X˜

text: X˜

text: and

text: therefore

text: deﬁne

text: a

text: weighted

text: branching

text: tree.

text: N˜

text: Recall

text: from

text: the

text: beginning

text: of

text: the

text: section

text: that

text: S˜∅

text: S˜i

text: X˜i,

text: i

text: and

text: S˜

text: r

text: i

text: i

text: A˜r

text: p

text: We

text: will

text: show

text: that

text: E

text: rn

text: Uˆ

text: rn

text: U˜

text: as

text: n

text: r

text: r

text: To

text: this

text: end,

text: note

text: that

text: we

text: can

text: split

text: Uˆ

text: and

text: U˜

text: r

text: r

text: Uˆ

text: max

text: Sˆ

text: Sˆ+

text: max

text: r

text: i

text: i

text: r

text: r

text: i

text: Mrc

text: i

text: Mr


text: (cid:87) (cid:87) Proposition 6.5 Suppose that Assumption 3.1 (i)-(ii) is satisﬁed. Then, for ν˜ , the probability (cid:12) (cid:12) ≥ k → ∞ measure of kr=0 U˜r, νk, the probabi(cid:12)(cid:12)lity measure o(cid:12)(cid:12)f kr=0 Ur, any p 1 and any rn such that − l→im∞ E[N] E[N˜] (E[N])rn/p rn = 0, n we have that → → ∞ W (ν˜ , ν ) 0 n . p rn rn (cid:80) { } { } P(cid:80)roof. We start by constructing a coupling of ν˜ and ν . Let ξ ∈ and ζ ∈ be two rn rn i i U i i U independent sequences of i.i.d. Uniform(0,1) random variables. Let F (k) = k f (j), F (k) = n j=1 n k f(j) and set j=1 − 1 − 1 τ˜ = ∗ log ξ and τ = ∗ log ξ , i i i i λ λ n − − N˜ = F 1(ζ ) and N = F 1(ζ ), i n i i i − { ∈ R ≥ } where g 1(t) = inf x : g(x) t (this is the standard inverse transform construction). { } { } {∅} Now use the N˜ and the N to construct two branching trees according to A˜ = = A and { ∈i ≤ i ≤ } { ∈ ≤ ≤ } 0 ≥ 0 A˜ = (i, i ) : i A˜ − , 1 i N˜ , A = (i, i ) : i A − , 1 i N for r 1. It remains r r r 1 r i r r r ∼1 r i to construct the sequences of service requirements. For the weighted branching tree we sample { } i.i.d. random variables χ ∈ having distribution B, independent of all other random variables, − i i U∈ − and set X˜ = χ τ˜ for each i U. Then let X = χ τ . i i i i i i To analyze the diﬀerence between the corresponding processes U˜ and U , we start by deﬁning the r r ∈ (cid:54) notion of a miscoupling. We say that there has been a miscoupling at node i A if N = N˜ and ≤ { ∈ ≤r i } i N | = N˜ | for all 1 k < r. Next, deﬁne C = i A : N | = N˜ | for all 1 k < r , which i k i k r r i k i k corresponds to the set of individuals in both trees that have no miscouplings along their paths. Following the same steps usedin the proof of Proposition 6.4, and with some abuse of notation, split the paths in U˜ and U as foll(cid:95)ows: (cid:95)  (cid:110) (cid:111) r r   (cid:44) U˜ = max S˜+, S˜+ max U˜(1), U˜(2) , r i i r r i∈A˜(cid:95)r∩Cr i∈A˜(cid:95)r∩Crc  (cid:110) (cid:111) and   (cid:44) U = max S+, S+ max U(1), U(2) . r i i r r ∈ ∩ ∈ ∩ i Ar Cr i Ar Crc By using Lemma 6.3 we obtain 30 ##LTLine## ##LTLine##

text: Proposition

text: Suppose

text: that

text: Assumption

text: (i)-(ii)

text: is

text: satisﬁed.

text: Then,

text: for

text: ν˜

text: the

text: probability

text: k

text: measure

text: of

text: U˜r,

text: νk,

text: the

text: measure

text: Ur,

text: any

text: p

text: and

text: any

text: rn

text: such

text: that

text: l→im∞

text: E[N]

text: E[N˜]

text: (E[N])rn/p

text: rn

text: n

text: we

text: have

text: that

text: W

text: (ν˜

text: ν

text: n

text: p

text: rn

text: rn

text: We

text: start

text: by

text: constructing

text: a

text: coupling

text: of

text: ν˜

text: and

text: ν

text: Let

text: ξ

text: and

text: ζ

text: be

text: two

text: rn

text: rn

text: i

text: i

text: U

text: i

text: i

text: U

text: independent

text: sequences

text: of

text: i.i.d.

text: random

text: variables.

text: Let

text: F

text: (k)

text: k

text: f

text: (j),

text: F

text: (k)

text: n

text: n

text: k

text: f(j)

text: and

text: set

text: τ˜

text: log

text: ξ

text: and

text: τ

text: log

text: ξ

text: i

text: i

text: i

text: i

text: λ

text: λ

text: n

text: N˜

text: F

text: and

text: N

text: F

text: i

text: n

text: i

text: i

text: i

text: R

text: where

text: g

text: inf

text: x

text: g(x)

text: t

text: (this

text: is

text: the

text: standard

text: inverse

text: transform

text: construction).

text: Now

text: use

text: the

text: N˜

text: and

text: the

text: N

text: to

text: construct

text: two

text: branching

text: trees

text: according

text: to

text: A˜

text: A

text: and

text: ∈i

text: i

text: A˜

text: (i,

text: i

text: i

text: A˜

text: i

text: N˜

text: A

text: (i,

text: i

text: i

text: A

text: i

text: N

text: for

text: r

text: It

text: remains

text: r

text: r

text: r

text: r

text: i

text: r

text: r

text: r

text: ∼1

text: r

text: i

text: to

text: construct

text: the

text: sequences

text: of

text: service

text: requirements.

text: For

text: the

text: weighted

text: branching

text: tree

text: we

text: sample

text: i.i.d.

text: random

text: variables

text: χ

text: having

text: distribution

text: B,

text: independent

text: of

text: all

text: other

text: random

text: variables,

text: i

text: i

text: U∈

text: and

text: set

text: X˜

text: χ

text: τ˜

text: for

text: each

text: i

text: U.

text: Then

text: let

text: X

text: χ

text: τ

text: i

text: i

text: i

text: i

text: i

text: i

text: To

text: analyze

text: the

text: diﬀerence

text: between

text: the

text: corresponding

text: processes

text: U˜

text: and

text: U

text: we

text: start

text: by

text: deﬁning

text: the

text: r

text: r

text: notion

text: of

text: a

text: miscoupling.

text: We

text: say

text: that

text: there

text: has

text: been

text: a

text: miscoupling

text: at

text: node

text: i

text: A

text: if

text: N

text: N˜

text: and

text: ≤r

text: i

text: i

text: N

text: N˜

text: for

text: all

text: k

text: r.

text: Next,

text: deﬁne

text: C

text: i

text: A

text: N

text: N˜

text: for

text: all

text: k

text: r

text: which

text: i

text: k

text: i

text: k

text: r

text: r

text: i

text: k

text: i

text: k

text: corresponds

text: to

text: the

text: set

text: of

text: individuals

text: in

text: both

text: trees

text: that

text: have

text: no

text: miscouplings

text: along

text: their

text: paths.

text: Following

text: the

text: same

text: steps

text: usedin

text: the

text: proof

text: of

text: Proposition

text: and

text: with

text: some

text: abuse

text: of

text: notation,

text: split

text: the

text: paths

text: in

text: U˜

text: and

text: U

text: as

text: r

text: r

text: U˜

text: max

text: S˜+,

text: S˜+

text: max

text: r

text: i

text: i

text: r

text: r

text: and

text: U

text: max

text: S+,

text: S+

text: max

text: r

text: i

text: i

text: r

text: r

text: i

text: Ar

text: Cr

text: i

text: Ar

text: Crc

text: By

text: using

text: Lemma

text: we

text: obtain

text: ##LTLine##

text: ##LTLine##


text: (cid:12) (cid:12) (cid:12) (cid:12) | | (cid:12) (cid:12) where Yrn is an Erlang ra(cid:12)(cid:12)ndom var(cid:12)(cid:12)iable(cid:12)(cid:12) with paramet(cid:12)(cid:12)ers (rn, 1). Since E[ Arn ] = (E[N])rn, (cid:12) (cid:12) (cid:12) − (cid:12) | − | 1 − 1 E[N] E[N˜] ≤ E[N] E[N˜] ∗ ∗ = , λ λn (cid:90) λE[N]E[Nˆ] λ and ∞ − − E[Y p ] = xrn+p−1e x dx = Γ(rn + p), rn (r 1)! Γ(r ) 0 n n where Γ(t) is the gamma function, we have that (cid:34)(cid:12) (cid:12) (cid:35) (cid:12)(cid:95) (cid:95) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) rn − rn (cid:12)p ≤ |E(cid:16)[N(cid:12) ] − E[N˜]|p (cid:12) Γ(r +(cid:17)p) E U˜(1) U(1) (cid:12) (E(cid:12) [N])rn n r r (cid:12) λp (cid:12) Γ(r ) n r=0 r=0 − p = O E[N] E[N˜] (E[N])rn rp (6.10) n → ∞ ∈ R as n , where in the last step we used that lim →∞ Γ(k)kα/Γ(k + α) = 1 for any α . k (cid:32) (cid:34)(cid:12) (cid:12) (cid:35)(cid:33) Combining (6.9) a(cid:12)n(cid:95)d (6.10) (cid:95)with ((cid:12)6.7) and (6.8(cid:16)), gives (cid:12) (cid:12) (cid:17) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) rn − rn p 1/p E − E U˜ U = O 1/p + E[N] E[N˜] (E[N])rn/p r , r r n n (cid:12) (cid:12) r=0 r=0 (cid:12) (cid:12) (cid:12) (cid:12) where − → E[N] E[N˜] (E[N])rn/p r 0 n → ∞ E → (cid:88) as n by assumption. To see that 0 as well, note that n E ρ˜ mn (cid:54) | ≤ (cid:54) | = β kP (N = k N˜ = k)f (k) ρ˜ max P (N = k N˜ = k). n E[N˜] n β 1≤k≤mn k=1 → Since ρ˜β ρβ < 1, it only re(cid:90)mai(cid:8)ns to{sho∈w that the−maximum} on the righ(cid:9)t hand side co≤nver≤ges to zero. To see this is the case let R = u (0, 1) : F 1(u) = k and note that for any 1 k m , k n n (cid:54) | (cid:90) − ≤ − − ≥ P (N = k N˜ = k) = (cid:8)1(F 1(u) k 1) + 1(F 1(u) k + 1) du (cid:9) R k (cid:90) − ≤ − − − ≥ − = 1(F 1(u) F 1(u) 1) + 1(F 1(u) F 1(u) + 1) du n n R k (cid:90) | − − − | ≥ = 1( F 1(u) F 1(u) 1) du n R k ≤ 1 | − − − | F 1(u) F 1(u) du, n 0 where the last integral is the Wasserstein distance of order one (W ) between distributions f and ⇒ 1 → n f (see, e.g., [21, 30]), which converges to zero since f f and E[N˜] E[N] (by Assumption 3.1 n 33 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: Yrn

text: is

text: an

text: Erlang

text: with

text: (rn,

text: Since

text: E[

text: Arn

text: (E[N])rn,

text: E[N]

text: E[N˜]

text: E[N]

text: E[N˜]

text: λ

text: λn

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