A NOTE ON THE EXTREMAL PROCESS OF THE SUPERCRITICAL GAUSSIAN FREE FIELD

FREE FIELD

ALBERTO CHIARINI, ALESSANDRA CIPRIANI, AND RAJAT SUBHRA HAZRA

ABSTRACT. We consider both the infinite-volume discrete Gaussian Free Field (DGFF) and the DGFF with zero boundary conditions outside a finite box in dimension larger or equal to 3. We show that the associated extremal process converges to a Poisson point process.

The result follows from an application of the Stein-Chen method fromArratia et al.(1989).

1. INTRODUCTION

In this article we study the behavior of the extremal process of the DGFF in dimension larger or equal to 3. This extends the result presented inChiarini et al.(2015) in which the convergence of the rescaled maximum of the infinite-volume DGFF and the 0-boundary condition field was shown. It was proved there that the field belongs to the maximal domain of attraction of the Gumbel distribution; hence, a natural question that arises is that of describing more precisely its extremal points. In dimension 2, this was carried out by Biskup and Louidor (2013, 2014) complementing a result of Bramson et al. (2013) on the convergence of the maximum; namely, the characterization of the limiting point pro- cess with a random mean measure yields as by-product an integral representation of the maximum. The extremes of the DGFF in dimension 2 have deep connections with those of Branching Brownian Motion (A¨ıd´ekon et al.(2013),Arguin et al.(2011,2012,2013)). These works showed that the limiting point process is a randomly shifted decorated Poisson point process, and we refer toSubag and Zeitouni(2015) for structural details. Ind ě 3, one does not get a non-trivial decoration but instead a Poisson point process analogous to the extremal process of independent Gaussian random variables. To be more precise, we letE :“ r0, 1s^dˆ p´8, `8sandVN :“ r0, n´1s^dXZ^dthe hypercube of volumeN “n^d. Letpϕ_αq_αPZd be the infinite-volume DGFF, that is a centered Gaussian field on the square lattice with covariancegp¨, ¨q, wheregis the Green’s function of the simple random walk.

We define the following sequence of point processes onE:

ηnp¨q:“ ÿ

αPV_N

ε^´α n,^ϕα´bN

aN

¯p¨q (1)

The first author’s research is supported by RTG 1845.

1

arXiv:1505.05324v1 [math.PR] 20 May 2015

where εxp¨q, x P E, is the point measure that gives mass one to a set containing x and zero otherwise, and

b_N :“

b gp0q

«

a2 logN´log logN`logp4πq 2a

2 logN ff

, a_N :“ gp0qpb_Nq^´1. (2) Heregp0qdenotes the variance of the DGFF. Our main result is

Theorem 1. For the sequence of point processesηn defined in(1)we have that ηn d

Ñη,

as n Ñ `8, whereη is a Poisson random measure on E with intensity measure given bydtb

`e^´zdz˘

wheredtbdz is the Lebesgue measure on E, andÑ^d is the convergence in distribution onM_ppEq^a.

The proof is based on the application of the two-moment method ofArratia et al.(1989) that allows us to compare the extremal process of the DGFF and a Poisson point process with the same mean measure. To prove that the two processes converge, we will exploit a classical theorem by Kallenberg.

It is natural then to consider also convergence for the DGFFpψ_αq_αPZd with zero bound- ary conditions outsideV_N. For the sequences of point measures

ρnp¨q:“ ÿ

αPV_N

ε^´_α

n,^ψα´bN

aN

¯p¨q (3)

we establish the following Theorem:

Theorem 2. For the sequence of point processesρn defined in(3)we have that ρn d

Ñη, as nÑ `8inM_ppEq, whereη is as in Theorem1.

The convergence is shown by reducing ourselves to check the conditions of Kallen- berg’s Theorem on the bulk of V_N, where we have a good control on the drift of the conditioned field, and then by showing that the process on the whole of V_N and on the bulk are close asnbecomes large.

The outline of the paper is as follows. In Section2we will recall the definition of DGFF and the Stein-Chen method, while Section 3 and Section4 are devoted to the proofs of Theorems1and2respectively.

aM_ppEq denotes the set of (Radon) point measures on E endowed with the topology of vague convergence.

2. PRELIMINARIES

2.1. The DGFF. Letdě3 and denote with} ¨ }the`₈-norm onZ^d. Letψ“ pψαq_αPZd be a discrete Gaussian Free Field with zero boundary conditions outside Λ Ă Z^d . On the spaceΩ :“ R^Zd endowed with its product topology, its lawPr_Λ can be explicitly written as

Pr_Λpdψq “ 1 Z_Λ exp

¨

˝´ 1 2d

ÿ

α,βPZ^d:}α´β}“1

`

ψα´ψβ

˘₂

˛

‚ ź

αPΛ

dψα

ź

αPZ^dzΛ

ε0pψαq.

In other words ψα “ 0 rP_Λ-a. s. if α P Z^dzΛ, and pψαq_αPΛ is a multivariate Gaussian random variable with mean zero and covariancepg_Λpα, βqq_α,βPZd, whereg_Λis the Green’s function of the discrete Laplacian problem with Dirichlet boundary conditions outsideΛ.

For a thorough review on the model the reader can refer for example toSznitman(2012). It is known (Georgii,1988, Chapter 13) that the finite-volume measureψadmits an infinite- volume limit asΛ Ò Z^d in the weak topology of probability measures. This field will be denoted as ϕ “ pϕ_αq_αPZd. It is a centered Gaussian field with covariance matrixgpα, βq forα, β P Z^d. With a slight abuse of notation, we write gpα´βqfor gp0, α´βqand also g_Λpαq “ g_Λpα, αq. gadmits a so-called random walk representation: ifPαdenotes the law of a simple random walkSstarted atαPZ^d, then

gpα, βq “ Eα

« ÿ

ně0

1tSn“βu

ff .

In particular this gives gp0q ă `8 for d ě 3. A comparison of the covariances in the infinite and finite-volume is possible in thebulkofVN: forδą0 this is defined as

V_N^δ :“!

α PV_N : }α´β} ąδn, @βPZ^dzV_N)

. (4)

In order to compare covariances in the finite and infinite-volume field, we recall the fol- lowing Lemma, whose proof is presented inChiarini et al.(2015, Lemma 7)).

Lemma 3. For anyδ ą0andα, βP V_N^δ one has gpα,βq ´C_d´

δN^1{d¯2´d

ď g_VNpα,βq ď gpα,βq. (5) In particular we have, g_VNpαq “ gp0q´

1`O´

N^p2´dq{d¯¯

uniformly forα PV_N^δ.

2.2. The Stein-Chen method. As main tool of this article we will use (and restate here) a theorem from Arratia et al.(1989). Consider a sequence of Bernoulli random variables pXαq_αPI whereXα „ BeppαqandI is some index set. For eachαwe define a subsetBα ĎI

which we consider a “neighborhood” of dependence for the variable Xα, such thatXα is nearly independent from X_β ifβP IzBα. Set

b₁ :“ ÿ

αPI

ÿ

βPB_α

pαp_β,

b₂:“ ÿ

αPI

ÿ

α‰βPB_α

E“

XαX_β‰ ,

b₃ :“ ÿ

αPI

Er|ErX_α´p_α|H₁s|s where

H₁ :“σ`

X_β : βPIzBα

˘.

Theorem 4 (Arratia et al. (1989, Theorem 2)). Let I be an index set. Partition the index set I into disjoint non-empty setsI_{1, . . . ,} I_{k. For anyα} P I_{, letpXα}q_αPI be a dependent Bernoulli process with parameter pα. LetpYαq_αPI be independent Poisson random variables with intensity p_α. Also let

Wj :“ ÿ

αPI_j

Xα and Zj :“ ÿ

αPI_j

Yα and λj :“ErWjs “ErZjs.

Then

}LpW1, . . . ,W_kq ´LpZ1, . . . ,Z_kq}TV ď2 min

!

1, 1.4` minλj

˘_´1{2)

p2b1`2b2`b3q (6) where} ¨ }TV denotes the total variation distance andLpW₁, . . . ,W_kqdenotes the joint law of these random variables.

3. PROOF OF THEOREM 1: THE INFINITE-VOLUME CASE

Proof. We recall that E “ r0 , 1s^dˆ p´8,`8s and V_N “ r0, n´1s^dXZ^d. To show the convergence ofηn toη, we will exploit Kallenberg’s theorem (Kallenberg,1983, Theorem 4.7). According to it, we need to verify the following conditions:

i) for any A, a bounded rectangle^binr0, 1s^d, andR “ px,ys Ă p´8,`8s Erη_npAˆ px,ysqs Ñ ErηpAˆ px,ysqs “ |A|pe^´x´e^´yq.

We adopt the convention e^´8 “ 0 and the notation|A|for the Lebesgue measure of A.

bAbounded rectanglehas the formJ1ˆ ¨ ¨ ¨ ˆJ_dwithJi“ r0, 1s X pai,bis,ai,bi PRfor all 1ďiďd.

ii) For allk ě 1, and A₁, A₂, . . . , A_k disjoint rectangles inr0, 1s^d and R₁, R₂, . . . , R_k, each of which is a finite union o disjoint f intervals of the typepx, ys Ă p´8,`8s,

PpηnpA₁ˆR₁q “ 0, . . . , ηnpA_kˆR_kq “ 0q

ÑPpηpA1ˆR1q “0, . . . , ηpAkˆRkq “0q “ exp

¨

˝´

k

ÿ

j“1

|Aj|ω` Rj

˘

˛

‚ (7) where ωpdzq:“e^´zdz.

Let us denote byu_Npzq:“a_Nz`b_N. The first condition follows by Mills ratio ˆ

1´ 1 t²

˙e^´t2{2

?2πt ďPpNp0, 1q ą tq ď e^´t2{2

?2πt, tą0. (8)

More precisely

Erη_npAˆ px,ysqs “ ÿ

αPnAXV_N

Ppϕ_α P puNpxq,uNpyqsq

ď ÿ

αPnAXV_N

¨

˚

˝ e^´uN

pxq2 2gp0q

?2πuNpxq´ e^´uN

pyq2 2gp0q

?2πuNpyq ˆ

1´ 1 uNpyq²

˙

˛

‹

‚ (9)

ď |nAXV_N|

˜e^´x`op1q

N ´e^´y`op1q N

ˆ

1´ 1

2gp0qlogNp1`op1qq

˙¸

Ñ |A|pe^´x´e^´yq. (10)

Similarly, one can plug in (9) the reverse bounds of (8) to prove the lower bound, and thus condition i).

To show ii), we need a few more details. Letk ě 1, A₁, . . . , A_k and R₁, . . . ,R_kbe as in the assumptions. Let us denote byI_j “nA_jXVN andI “I₁Y. . .YI_k. ForαP I_jdefine

X_α :“1^!ϕα´bN aN PR_j

)

and p_α :“P`

pϕ_α´b_Nq{a_N P R_j˘

. Choose now a smalle ą0 and fix the neighborhood of dependence Bα :“ B`

α,plogNq^2`2e˘

XI forα P I. Let Wj :“ ř

αPI_jXα and Zj be as in Theorem4.

By the simple observation that

PpηnpA₁ˆR₁q “ 0, . . . ,ηnpA_kˆR_kq “ 0q “PpW₁ “0, . . . , W_k “0q,

to prove the convergence (7), we can use Theorem4 and show that the error bound on the RHS of (6) goes to 0.

First we bound b1 as follows. By definition of R1, R2, . . . , R_k, there exists z P Rsuch thatRj Ă pz,`8sfor 1ď jďk. Hence for any 1ďj ďk, for anyα PI_j we have that

pα “P

ˆϕα´bN

a_N P R_j

˙

ďPpϕα ąuNpzqq⁽⁸⁾ď e^´uN

pzq2 2gp0q

?2πuNpzq b

gp0q.

The bound is independent ofαand j, therefore for someCą0

b1ďCNplogNq^dp2`2eqe^´2zN^´2Ñ0. (11) Forb2note that it was shown inChiarini et al.(2015) that forz PRandα ‰βPVN

Ppϕ_α ąu_Npzq, ϕβ ąu_Npzqq ď p2´κq^3{2

κ^1{2 N^´2{p2´κqmax!

e^´2z1tzď0u, e^´2z{p2´κq1tzą0u

) . (12) Here we have introduced κ :“ P0

´

Hr₀ “ `8

¯

P p0, 1q and Hr₀ “ inftně1 : Sn “0u.

Observe that for any 1ď jďk,α P I _andβP Bαone has

ErXαX_βs ďPpϕα ąuNpzq,ϕ_β ąuNpzqq so that by (12) we can find some constantC¹ ą0 such that

b2 ďC¹N^´κ{p2´κqplogNq^dp2`2eqmax

!

e^´2z1tzď0u, e^´2z{p2´κq1tzą0u

) Ñ0.

Finally we need to handle b3. From Section2.2 we set forα P I,H₁ :“σ`

X_β : βPIzBα

˘ and we defineH₂ :“σ`

ϕβ : βP IzBα

˘. We observe that b₃ “ ÿ

αPI

Er|ErX_α´p_α|H₁s|s ď ÿ

αPI

Er|ErX_α|H₂s ´p_α|s

sinceH₁ĎH₂and using the tower property of the conditional expectation. Now denote byUα :“ Z^dz pIzBαq. Let us abbreviateu_NpR_jq :“ tu_Npyq : y P R_ju. Then for α P I_j and 1 ď j ď k, by the Markov property of the DGFF (Rodriguez and Sznitman, 2013, Lemma 1.2) we have that

ErXα|H₂s “Pr_U

αpψ_α`µα P uNpRjqq P´a. s.

wherepψαq_αPZd is a Gaussian Free Field with zero boundary conditions outsideUα and µα “

ÿ

βPIzB_α

Pα

´

HIzB_α ă `8, S_HIzB

α “β

¯ ϕ_β.

Here H_Λ :“ inftně0 : Sn PΛu, Λ ĂZ^d. Now as inChiarini et al. (2015) one can show, using the Markov property, that

Varrµαs ď sup

βPIzB_α

gpα,βq ď c

plogNq2p1`eqpd´2q

for somec ą0. Hence we get that there exists a constant c¹ ą0 (independent ofα and j) such that

P

´

|µα| ą puNpzqq^´1´e

¯

ďc¹exp

´

´plogNqp2d´5qp1`eq¯

. (13)

Recalling thatR_j Ă pz, `8sfor all 1ďj ďk, this immediately shows that fordě3

k

ÿ

j“1

ÿ

αPI_j

E

”ˇ ˇ ˇrP_U

αpψα`µα PuNpRjqq ´pα

ˇ ˇ

ˇ1t^|µα|ąpu_Npzqq^´1´eu ı

Ñ0.

So to show thatb3 Ñ0 we are left with proving ÿk

j“1

ÿ

αPI_j

E

”ˇ ˇ

ˇPr_Uαpψα`µα P uNpR_jqq ´pα

ˇ ˇ

ˇ1t^|µα|ďpu_Npzqq^´1´eu ı

Ñ0. (14)

We now focus on the term inside the summation. For this, first we writeR_j “Ť_m

l“1pw_l,r_ls with ´8 ă w1 ăr1 ă w2 ă ¨ ¨ ¨ ă rm ď `8for somem ě 1. Hence, we can expand the difference in the absolute value of (14) as follows:

´

p_α´Pr_U

αpψ_α`µα P u_NpR_jqq

¯

“ ÿm l“1

´

Ppϕα P puNpw_lq,uNpr_lqsq ´rP_U

αpψα`µα P puNpw_lq,uNpr_lqsq

¯

“

m

ÿ

l“1

´

Ppϕ_α ąu_Npw_lqq ´rP_Uαpψ_α`µα ąu_Npw_lqq

¯

´ ÿm l“1

´

Ppϕα ąuNpr_lqq ´rP_U

αpψα`µα ąuNpr_lqq

¯

(15) (if r_l “ `8for some l, we conventionally setPpϕ_α ą uNpr_lqq “ 0 and similarly for the other summand). Using the triangular inequality in (14), it turns out that to finish it is enough to show that for an arbitrarywPR,

ÿ

αPI

E

”ˇ ˇ ˇPr_U

αpψα`µα ąuNpwqq ´Ppϕα ąuNpwqq ˇ ˇ

ˇ1t^|µα|ďpu_Npzqq^´1´eu ı

Ñ0. (16) For this, first we show that onQ :“

!

Ppϕα ąuNpwqq ąPr_U

αpψα`µα ąuNpwqq )

ÿ

αPI

E

”´

Ppϕ_α ąu_Npwqq ´Pr_Uαpψα`µα ąu_Npwqq

¯1t^|µα|ďpuNpzqq^´1´eu1Q

ı

Ñ0. (17)

This follows from the same estimates ofT1,2and Claim 6 ofChiarini et al.(2015). Indeed onQX

!

|µα| ď pu_Npzqq^´1´e) ÿ

αPI

pPpϕ_α ąuNpwqq ´P_U

αpψ_α`µα ąuNpwqqq

ď ÿ

αPI

agp0qe^´uN

pwq2 2gp0q

?2πuNpwq

¨

˚

˚

˝1´ p1`op1qq

¨

˚

˚

˝

agU_αpαquNpwqe

ˆ 1´ ^gp0q

gUαpαq

˙uNpwq2 2gp0q `op1q

agp0quNpwqp1`op1qq

˛

‹

‹

‚

˛

‹

‹

‚

ďCN

agp0qe^´uN

pwq2 2gp0q

?2πu_Npwq op1q “ op1q.

Similarly one can show that on the complementary event Q^c (recall (17) for the defini- tion ofQ₎

ÿ

αPI

E

”´

Pr_U

αpψα`µα ąuNpwqq ´Ppϕα ąuNpwqq

¯1t^|µα|ďpu_Npzqq^´1´eu1Q^c

ı

“op1q. This shows thatb3Ñ0. Hence from Theorem4it follows that

ˇ ˇ ˇ ˇ ˇ ˇ

PpW₁“0, . . . ,W_k “0q ´ źk j“1

P`

Z_j “0˘ ˇ ˇ ˇ ˇ ˇ ˇ

“op1q,

having used the independence of theZj’s. Notice that by definitionZjis a Poisson random variable with intensityř

αPI_jP`

pϕ_α´b_Nq{a_N P R_j˘

. DecomposingR_j as a union of finite intervals and using Mills ratio, similarly to the argument leading to (10), one has

PpZ_j“0q Ñexpp´|A_j|ωpR_jqq (recallωpRjq “ ş

R_je^´zdz). Hence it follows that źk

j“1

PpZ_j “0q Ñexp

¨

˝´ ÿk j“1

|A_j|ωpR_jq

˛

‚, (18)

which completes the proof of ii) and therefore of Theorem1.

4. PROOF OFTHEOREM 2: THE FINITE-VOLUME CASE

We will now show the theorem for the field with zero boundary conditions. As re- marked in the Introduction, since on the bulk defined in (4) we have a good control on the conditioned field, we will first prove convergence therein, and then we will use a converging-together theorem to achieve the final limit. We will first need some notation used throughout the Section: first, we considerpψ_αq_αPVN with lawPr_N :“Pr_VN. We also use

the shortcut g_Np¨, ¨q “ g_VNp¨, ¨q. We will need the notation C_K<sup>`</sup>pEqfor the set of positive, continuous and compactly supported functions on E“ r0, 1s<sup>d</sup>ˆ p´8,`8s.

F^IGURE1. Sketch of the proof of Theorem2

We first begin with a lemma on the point process convergence on bulk. Define a point process onEby

ρ^δ_np¨q “ ÿ

αPV_N^δ

ε^´α n,^ψα´bN

aN

¯p¨q. (19)

Lemma 5. Letδ ą0. OnM_ppEq,ρ^δ_n Ñ^d ρ^δ whereρ^δis a Poisson random measure with intensity dt|

rδ,1´δsd b`

e^´xdx˘_c .

Proof. We will show i) and ii) of Page 4 (and from which we will borrow the notation starting from now).

cdt_|

rδ,1´δsd is the restriction of the Lebesgue measure torδ, 1´δs^d.

i) We begin with an upper bound onEr_N“

ρ^δ_npAˆ px,ysq‰ : ÿ

αPnAXV_N^δ

Pr_Npψα ąuNpxqq ´rP_Npψα ąuNpyqq

(8)

ď ÿ

αPnAXV_N^δ

e^´uN

pxq2 2gNpαq

?2πuNpxq b

gNpαq ´ e^´uN

pyq2 2gNpαq

?2πuNpyq b

gNpαq p1`op1qq

Lemma3

“

ÿ

αPnAXV_N^δ

e^{´ uN}

pxq2 2gp0qp1`cnq

?2πuNpxq b

gp0qp1`cnq ´ e^{´ uN}

pyq2 2gp0qp1`cnq

?2πuNpyq b

gp0q p1`cnq

nÑ`8ÝÑ pe^´x´e^´yq ˇ ˇ

ˇAX rδ, 1´δs^d ˇ ˇ

ˇ. (20)

We stress that in the second step the error term cn :“ O` n^2´d˘

coming from Lemma 3 guarantees the convergence in the last line. The lower bound follows similarly.

ii) To show the second condition we again use Theorem4. Let A₁, . . . ,A_k and R₁, . . . ,R_k be as in proof of Theorem1. LetI_j :“ nAjXV_N^δ and I “ I₁Y ¨ ¨ ¨ YI_k. For e ą0 we are setting Bα :“ B

´

α, plogNq^2p1`eq

¯

XI. Note that, albeit slightly different, we are using the same notations for the neighborhood of dependence and the index sets of Section 3, but no confusion should arise. Observe that there existsz PRsuch that for all 1ď j ďk, R_jĂ pz,8s; we have

pα “rP_N

ˆψα´bN

aN

P uNpRjq

˙

ďPr_Npψ_α ąuNpzqq⁽⁸⁾ď e^´uN

pzq2 2gp0q

?2πuNpzq b

gp0q

where we have also used the fact that gNpαq ď gp0q. The bound on b1 (cf. Theorem 4) follows exactly as in (11) and yields that, for someCą0,

b₁ďCNplogNq^dp2`2eqe^´2zN^´2Ñ0.

The calculation ofb2can be performed similarly using the covariance matrix of the vector pψα, ψ_βq, α ‰ β P V_N^δ and Lemma 3. This gives that for someC, C¹ ą 0 independent of α,βPV_N^δ

b₂ ď ÿ

αPI

ÿ

βPB_α

C logNexp

ˆ

´ u_Npzq² gp0q `gpα´βq

´

1`O´

N^p2´dq{d¯¯˙

ďC¹N^´κ{p2´κqplogNq^2dp1`eqmax

!

e^´2z1tzď0u, e^´2z{p2´κq1tzą0u

) Ñ0

(cf. Chiarini et al. (2015)). We will now pass to b3. We repeat our choice of H₁ “ σ`

X_β : βPIzBα

˘andH₂“σ`

ψ_β : βPIzBα

˘so thatb3becomes

k

ÿ

j“1

ÿ

αPI_j

Er_N

”ˇ ˇ

ˇrE_NrXα´pα|H₁s ˇ ˇ ˇ ı

ď

k

ÿ

j“1

ÿ

αPI_j

rE_N

”ˇ ˇ

ˇrE_NrXα|H₂s ´pα

ˇ ˇ ˇ ı

. We defineUα :“VNzpIzBαq. By the Markov property of the DGFF

rE_NrXα|H₂s “Pr_U

αpξα`hα P uNpRjqq Pr_N´a.s. (21) forpξαq_αPZd a DGFF with lawPr_U

α and phαq_αPZd is independent ofξ. FromChiarini et al.

(2015) we can see that, forαP V_N^δ andNlarge enough such thatB

´

α, plogNq^2p1`eq

¯

ĹVN, Varrhαs “ ÿ

βPIzB_α

Pα

´

H_IzBα ă `8, S_HIzB

α “ β

¯

g_Npα, βq

ď sup

βPIzBα

gNpα, βq ď c

plogNq2p1`eqpd´2q. This yields

k

ÿ

j“1

ÿ

αPI_j

rE_N

”ˇ ˇ ˇPr_U

αpξ_α`hαq ą uNpRjqq ´pα

ˇ ˇ

ˇ1t^|hα|ąpu_Npzqq^´1´eu ı

Ñ0. (22)

It then suffices to show

k

ÿ

j“1

ÿ

αPI_j

rE_N

”ˇ ˇ ˇPr_U

αpξ_α`hαq ą uNpRjqq ´pα

ˇ ˇ

ˇ1t^|hα|ďpu_Npzqq^´1´eu ı

Ñ0. (23)

One sees that the breaking up (15) can be performed also here replacingϕα andψα (with their laws) with ψα andξα (with their laws) respectively, andµα with hα. Accordingly, it is enough to show that

ÿ

αPI

rE_N

”ˇ ˇ ˇPr_U

αpξ_α`hα ąuNpwqq ´Pr_Npψ_α ąuNpwqq ˇ ˇ

ˇ1t^|hα|ďpu_Npzqq^´1´eu ı

Ñ0 (24) for allwPR. To this aim, we choose for anywPRthe event

Q¹:“

!

rP_Npψα ąuNpwqq ąPr_Uαpξα`hα ąuNpwqq )

and we proceed as in (17) with the help of Lemma3to show (24). Given this, the conver- genceb3Ñ0 is finally ensured. Hence we can conclude that

}LpW1, . . . ,W_kq ´LpZ1, . . . , Z_kq}_TV Ñ0

where Z_jare i. i. d. Poisson of mean pα. By Mills ratio, as in (20) we see that PpZ_j “0q Ñ exp

´

´ ˇ ˇ

ˇA_jX rδ, 1´δs^d ˇ ˇ ˇωpR_jq

¯ .

From this it follows that the two conditions i) and ii) of Kallenberg’s Theorem are satisfied, and thus we obtain the convergence to a Poisson point process with mean measure given

in i).

Proof of Theorem2. M_ppEqis a Polish space with metric dp: dppµ, µ¹q “ ÿ

iě1

mint|µpf_iq ´µ¹pf_iq|, 1u

2ⁱ , µ, µ¹P M_ppEq

for a sequence of functions fiP C_K^`pEq(cf. Resnick(1987, Section 3.3)). Therefore we are in the condition to use a converging-together theorem (Resnick,2007, Theorem 3.5), namely to prove thatρn d

Ñη it is enough to show the following:

(a) ρ^δ_n Ñ^d ρ^δ, asnÑ `8.

(b) ρ^δ Ñ^d ηasδÑ0.

(c) For everyeą0,

limδÑ0 lim

nÑ`8Pr_N

´ dp

´ ρn,ρ^δ_n

¯ ąe

¯

“0. (25)

Note that by Lemma5, (a) is satisfied. For f P C_K^`pEq, the Laplace functional ofρ^δis given by (cf. Resnick(1987, Prop. 3.6))

Ψδpfq:“E

” exp

´

´ρ^δpfq

¯ı

“exp ˆ

´ ż

E

´

1´e^´fpt,xq

¯ dt_|

rδ,1´δsd e^´xdx

˙ .

Hence by the dominated convergence theorem we can exchange limit and expectation as δ Ñ0 to obtain that

Ψδpfq Ñ exp ˆ

´ ż

E

´

1´e^´fpt,xq

¯

dte^´xdx

˙

and the right hand side is the Laplace functional ofηat f. This shows (b).

Hence to complete the proof it is enough to show (25). Thanks to the definition of the metric dpit suffices to prove that for f P C_K^`pEqand fore ą0

lim sup

δÑ0

nÑ`8lim rP_N

´ˇ ˇ

ˇρnpfq ´ρ^δ_npfq ˇ ˇ ˇąe

¯

“0.

Without loss of generality assume that the support of f is contained inr0, 1s^dˆ rz0, `8q for some z0 P R. Choosing n large enough such that uNpz0q ą 0 and gNpαq ď gp0q, we

obtain that Er_N

”ˇ ˇ

ˇρnpfq ´ρ^δ_npfq ˇ ˇ ˇ ı

“Er_N

» –

ÿ

αPV_NzV_N^δ

f ˆα

n,ψα´bN

aN

˙

1^!ψα´_bN

aN ąz₀)

fi fl

ďsup

zPE

|fpzq| ÿ

αPVNzV_N^δ

Pr_N

ˆψα´bN

aN

ąz₀

˙ (8)

ďC ÿ

αPVNzV_N^δ

e^{´uNpz0q2{gp0q}

?2πu_Npz₀q b

gp0q

ďC¹

´

1´ p1´2δq^d

¯ e^´z0

asn Ñ `8for some positive constants C, C¹. Now letting δ Ñ 0 the result follows and this completes the proof.

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TECHNISCHEUNIVERSITAT¨ BERLIN, MA 766, STRASSE DES17. JUNI136, 10623 BERLIN, GERMANY

E-mail address:chiarini@math.tu-berlin.de

WEIERSTRASSINSTITUTE, MOHRENSTRASSE39, 10117 BERLIN, GERMANY

E-mail address:Alessandra.Cipriani@wias-berlin.de

THEORETICALSTATISTICS ANDMATHEMATICSUNIT, INDIANSTATISTICALINSTITUTE, 203, B.T. ROAD, KOLKATA, 700108, INDIA

E-mail address:rajatmaths@gmail.com