Optimal Berry-Esseen bounds on the Poisson space

Transcription

1 Optimal Berry-Esseen bounds on the Poisson space Ehsan Azmoodeh Unité de Recherche en Mathématiques, Luxembourg University Giovanni Peccati Unité de Recherche en Mathématiques, Luxembourg University July 7, 205 Abstract We establish new lower bounds for the normal approximation in the Wasserstein distance of random variables that are functionals of a Poisson measure. Our results generalize previous findings by Nourdin and Peccati 202, 205 and Biermé, Bonami, Nourdin and Peccati 203, involving random variables living on a Gaussian space. Applications are given to optimal Berry-Esseen bounds for edge counting in random geometric graphs. Keywords: Berry-Esseen Bounds; Limit Theorems; Optimal Rates; Poisson Space; Random Graphs; Stein s Method; U-statistics. MSC 200: 60H07; 60F05; 05C80. Contents Introduction. Overview Main abstract result and some preliminaries Preliminaries 3 2. Poisson measures and chaos Malliavin operators Some estimates based on Stein s method Proof of Theorem. 6 4 Applications to U-statistics 7 4. Preliminaries Geometric U-statistics of order Edge-counting in random geometric graphs Connection with generalized Edgeworth expansions 7 Azmoodeh is supported by research project FR-MTH-PUL-2PAMP.

2 Introduction. Overview Let, be a Borel space endowed with a σ-finite non-atomic measure µ, and let η be a compensated Poisson random measure on the state space,, with non-atomic and σ-finite control measure µ for the rest of the paper, we assume that all random objects are defined on a common probability space Ω, F, IP. Consider a sequence of centered random variables F n = F n η, n and assume that, as n, VarF n and F n converges in distribution to a standard Gaussian random variable. In recent years see e.g. [2, 6, 7, 8, 0, 6, 2, 23] several new techniques based on the interaction between Stein s method [4] and Malliavin calculus [9] have been introduced, allowing one to find explicit Berry-Esseen bounds of the type df n, N, n,. where d is some appropriate distance between the laws of F n and N, and { : n } is an explicit and strictly positive numerical sequence converging to 0. The aim of this paper is to find some general sufficient conditions, ensuring that the rate of convergence induced by in. is optimal, whenever d equals the -Wasserstein distance d W, that is: df n, N = d W F n, N := sup E[hF n ] E[hN],.2 h Lip with Lipa indicating the set of a-lipschitz mappings on IR a > 0. As usual, the rate of convergence induced by is said to be optimal if there exists a constant c 0, independent of n such that, for n large enough, d W F n, N c, ]..3 As demonstrated below, our findings generalize to the framework of random point measures some previous findings see [, 3, 3, 5] for random variables living on a Gaussian space. Several important differences between the Poisson and the Gaussian settings will be highlighted as our analysis unfolds. Important new applications U-statistics, in particular to edge-counting in random geometric graphs, are discussed in Section 4..2 Main abstract result and some preliminaries Let the above assumptions and notation prevail. The following elements are needed for the subsequent discussion, and will be formally introduced and discussed in Section 2.2: For every z and any functional F = F η, the difference or add-one cost operator D z F η = F η + δ z F η. For reasons that are clarified below, we shall write F dom D, whenever IE D sf 2 µds <. The symbol L denotes the pseudo-inverse of the generator of the Ornstein-Uhlenbeck semigroup on the Poisson space. We also denote by N N 0, a standard Gaussian random variable with mean zero and variance one. It will be also necessary to consider the family F W := {f : IR IR : f and f Lip2}, whereas the notation F 0 indicates the subset of F W that is composed of twice continuously differentiable functions such that f and f 2. For any two sequences {a n } n and {b n } n of non-negative real numbers, the notation a n b n indicates that lim n a n bn =. The next theorem is the main theoretical achievement of the present paper. 2

3 Theorem.. Let {F n : n } be a sequence of square-integrable functionals of η, such that IEF n = 0, and F n dom D. Let { : n } be a numerical sequence such that ϕ n + ϕ 2 n, where ϕ n := IE 2, DF n, DL F n L µ 2.4 ϕ 2 n := IE D z F n 2 D z L F n µdz..5 I For every n, one has the estimate d W F n, N. II Fix f F 0, set R f nz := 0 f F n + ud z F n u du for any z, and assume moreover that the following asymptotic conditions are in order : i a is finite for every n; b 0 as n ; and c there exists m such that > 0 for all n m. ii For µdz-almost every z, the sequence D z F n converges in probability towards zero. iii There exist a centered two dimensional Gaussian random vector N, N 2 with IEN 2 = IEN 2 2 =, and IEN N 2 = ρ, and moreover a real number α 0 such that F n, DF n, DL F n L 2 µ law N, αn 2. iv There exists a sequence {u n : n } of deterministic and non-negative measurable functions such that u nzµdz/ β <, and moreover { } D z F n 2 D z L F n R f nzµdz u n z R f L nzµdz Ω 0, and sup n +ɛ u nz +ɛ µdz <, for some ɛ > 0. Then, as n, we have IE f F n F n ff n { } β 2 + ρ α IEf N. III If Assumptions i iv at Point II are verified and ρα β 2, then the rate of convergence induced by is optimal, in the sense of.3. Remark.. It is interesting to observe that Assumptions II-ii and II-iv in the statement of Theorem. do not have any counterpart in the results on Wiener space obtained in [3]. To see this, let X denote a isonormal Gaussian process over a real separable Hilbert space H, and assume that {F n : n } is a sequence of smooth functionals in the sense of Malliavin differentiability of X for example, each element F n is a finite sum of multiple Wiener integrals. Assume that IEF 2 n =, and write := IE DF n, DL F n H 2. Assume that > 0 for all n and also that, as n, 0 and the two dimensional random vector F n, DFn, DL F n H converges in distribution to a centered two dimensional Gaussian vector N, N 2, such that IEN 2 = IEN 2 2 = and IEN N 2 = ρ 0. Then, the results of [3] imply that, for any function f F W, IE f F n F n ff n 3 ρief N,.6

4 where, as before, N N 0,. This implies in particular that the sequence determines an optimal rate of convergence, in the sense of.3. Also, on a Gaussian space one has that relation.6 extends to functions of the type f x, where f x is the solution of the Stein s equation associated with the indicator function { x} see Section 2.3 below: in 2 2π. this case the limiting value equals ρ 3 x2 x e 2 2 Preliminaries 2. Poisson measures and chaos As before,,, µ indicates a Borel measure space such that is a Borel space and µ is a σ- finite and non-atomic Borel measure. We define the class µ as µ = {B : µ B < }. The symbol η = { η B : B µ } indicates a compensated Poisson random measure on, with control µ. This means that η is a collection of random variables defined on the probability space Ω, F, IP, indexed by the elements of µ, and such that: i for every B, C µ such that B C =, η B and η C are independent, ii for every B µ, ηb has a centered Poisson distribution with parameter µ B. Note that properties i-ii imply, in particular, that η is an independently scattered or completely random measure. Without loss of generality, we may assume that F = σ η, and write L 2 IP := L 2 Ω, F, IP. See e.g. [9, 7] for details on the notions evoked above. Fix n. We denote by L 2 µ n the space of real valued functions on n that are square-integrable with respect to µ n, and we write L 2 s µ n to indicate the subspace of L 2 µ n composed of symmetric functions. We also write L 2 µ = L 2 µ = L 2 sµ. For every f L 2 sµ n, we denote by I n f the multiple Wiener-Itô integral of order n, of f with respect to η. Observe that, for every m, n, f L 2 sµ n and g L 2 sµ m, one has the isometric formula see e.g. [7]: IE[I n fi m g] = n! f, g L 2 µ n n=m. 2. The Hilbert space of random variables of the type I n f, where n and f L 2 sµ n is called the nth Wiener chaos associated with η. We also use the following standard notation: I f = η f, f L 2 µ; I 0 c = c, c R. The following proposition, whose content is known as the chaotic representation property of η, is one of the crucial results used in this paper. See e.g. [7]. Proposition 2. Chaotic decomposition. Every random variable F L 2 F, P = L 2 P admits a unique chaotic decomposition of the type F = EF + I n f n, 2.2 where the series converges in L 2 and, for each n, the kernel f n is an element of L 2 sµ n. 2.2 Malliavin operators We recall that the space L 2 IP; L 2 µ L 2 Ω, F, IP µ is the space of the measurable random functions u : Ω IR such that [ ] E u 2 z µdz <. In what follows, given f L 2 sµ q q 2 and z, we write fz, to indicate the function on q given by z,..., z q fz, z,..., z q. n 4

5 a The derivative operator D. The derivative operator, denoted by D, transforms random variables into random functions. Formally, the domain of D, written domd, is the set of those random variables F L 2 P admitting a chaotic decomposition 2.2 such that n nn! f n 2 L 2 µ n <. 2.3 If F verifies 2.3 that is, if F domd, then the random function z D z F is given by D z F = n ni n fz,, z. 2.4 Plainly DF L 2 IP; L 2 µ, for every F dom D. For every random variable of the form F = F η and for every z, we write F z = F z η = F η + δ z. The following fundamental result combines classic findings from [] and [8, Lemma 3.]. Lemma 2.. For every F L 2 IP one has that F is in dom D if and only if the mapping z F z F is an element of L 2 P; L 2 µ. Moreover, in this case one has that D z F = F z F almost everywhere dµ dip. b The Ornstein-Uhlenbeck generator L. The domain of the Ornstein-Uhlenbeck generator see e.g. [9, ], written dom L, is given by those F L 2 P such that their chaotic expansion 2.2 verifies n 2 n! f n 2 L 2 µ n <. n If F doml, then the random variable LF is given by LF = n ni n f n. 2.5 We will also write L to denote the pseudo-inverse of L. Note that ELF = 0, by definition. The following result is a direct consequence of the definitions of D and L, and involves the adjoint δ of D with respect to the space L 2 IP; L 2 µ see e.g. [9, ] for a proof. Lemma 2.2. For every F doml, one has that F domd and DF belongs to the domain to the adjoint δ of D. Moreover, δdf = LF Some estimates based on Stein s method We shall now present some estimates based on the use of Stein s method for the onedimensional normal approximation. We refer the reader to the two monographs [4, 4] for a detailed presentation of the subject. Let F be a random variable and let N N 0,, and consider a real-valued function h : IR IR such that the expectation E[hX] is well-defined. We recall that the Stein equation associated with h and F is classically given by hu IE[hF ] = f u ufu, u IR. 2.7 A solution to 2.7 is a function f depending on h which is Lebesgue a.e.-differentiable, and such that there exists a version of f verifying 2.7 for every x IR. The following lemma gathers together some fundamental relations. Recall the notation F W and F 0 introduced in Section.2. Lemma 2.3. i If h Lip, then 2.7 has a solution f h that is an element of F W. 5

6 ii If h is twice continuously differentiable and h, h, then 2.7 has a solution f h that is an element of F 0. iii Let F be an integrable random variable. Then, d W F, N = sup h Lip IE[fh F F f h F ] sup f F W IE[fF F f F ]. iv If, in addition, F is a centered element of dom D, then d W F, N IE DF, DL 2 F L µ IE D z F 2 D z L F µdz. Both estimates at Point i and ii follow e.g. from [5, Theorem.]. Point iii is an immediate consequence of the definition of d W, as well as of the content of Point i and Point ii in the statement. Finally, Point iv corresponds to the main estimate established in [6]. 3 Proof of Theorem. We start with a general lemma. Lemma 3.. Let F be such that IEF = 0 and F domd. Assume that N N 0,. For any f F 0, and z we set Then, R f z := IE f F F ff = IE + IE 0 f F + ud z F u du. 3. f F DF, DL F L µ D z F 2 D z L F R f zµdz. Proof. Using Lemma 2.2 and the characterization of δ as the adjoint of D, we deduce that, for any f F 0 IEF ff = IELL F ff = IEδ DL F ff = IE DfF, DL F L 2 µ. 3.2 In view of Lemma 2. and of a standard application of Taylor formula, one immediately infers that Fz D z ff := ff z ff = f F D z F + f uf z udu F 3.3 = f F D z F + D z F 2 Plugging 3.3 into 3.2, we deduce the desired conclusion. 0 f F + ud z F u du. Proof of Theorem.. Part I is a direct consequence of Lemma 2.3-iv. To prove Point II, we fix f F 0, and use Lemma 3. to deduce that IE f F n F n ff n = IE f F n DF n, DL F n L 2 µ + IE D z F n 2 D z L F n Rnzµdz f := I,n + I 2,n. 6

7 Assumption iii implies that sup IE n DFn, DL 2 F n L 2 µ < +. Since f by assumption, we infer that the class { f F n DF n, DL F n L 2 µ : n } is uniformly integrable. Assumption iii implies therefore that, as n, I,n IE f N αn 2 = ρ α IEf N. To deal with the term I 2,n, first note that for each z, Assumptions ii and iii and Slutsky Theorem imply that, for any u 0,, F n + ud z F n law N. Therefore, using the fact that f 2, and by a direct application of the dominated convergence Theorem, we infer that IER f nz 0 IEf Nudu = 2 IEf N. 3.4 At this point, Assumption iv and the triangle inequality immediately imply that, in order to obtain the desired conclusion, it is sufficient to prove that, as n, { u n z IER f nz } 2 IEf N µdz To show 3.5, it is enough to prove that the function integrated on the right-hand side is uniformly integrable: this is straightforward, since Rnz f 2 IEf N 2, and of the fact that the sequence n +ɛ u nz +ɛ µdz is bounded. In view of the first equality in Lemma 2.3-iii, to prove the remaining Point III in the statement, it is enough to show that there exists a function h such that h, h, and IE[f h N] 0. Selecting hx = sin x, we deduce from [, formula 5.2] that IEf h N = 3 IE[sinNH 3 N] = e /2 > 0, thus concluding the proof. 4 Applications to U-statistics 4. Preliminaries In this section we shall present some important application of our theoretical finding. We introduce the concept of a U-statistics associated with the Poisson random measure η. Those are the most natural examples of elements in L 2 IP and having a finite Wiener-Itô expansion. Definition 4.. Fix k. A random variable F is called a Ustatistics of order k, based on a Poisson random measure η with control measure µ, if there exists a symmetric kernel h L sµ k such that F = hx, 4. x η k where the symbol η k indicates the class of all k-dimensional vectors x = x,, x k such that x i η and x i x j for every i j k. 7

8 The following crucial fact on Poisson U-statistics is taken from [2, Lemma 3.5 and Theorem 3.6]. We shall also adopt the special notation L p,p k, µ k = L p k, µ k L p k, µ k for p, p. Proposition. Consider a kernel h L sµ k such that the corresponding U-statistic F given by 4.0 is square-integrable. Then, h is necessarily square-integrable, and F admits a representation of the form F = IEF + I i g i, where g i x i := h i x i = i= k hx i, x k i dµ k i, x i i, 4.2 i k i for i k, and g i = 0 for i > k. In particular, h = g k, and the projection g i L,2 s µ i for each i k. Now, let X be a compact subset of IR d endowed with the Lebesgue measure l, and let M be a locally compact space, that we shall call the mark space, endowed with a probability measure ν. We shall assume that X contains the origin in its interior, is symmetric, that is: X = X, and that the boundary of X is negligible with respect to Lebesgue measure. We set = X M, and we endow such a product space with the measure µ = l ν on IR d. Define µ n = nµ, and let η n be a Poisson measure on with control measure µ n. As before, h L,2 s k, µ k is such that each functional F n := x η k hx, n is a square-integrable U-statistic. In the terminology of [2] each F n is a geometric U-statistic. Note that, according to Proposition, we have where F n = IEF n + g i,n x i = n k i k i k F i,n := IEF n + i= k I i g i,n, i= k i hx i, x k i dµ k i := n k i h i x i, x i i, 4.3 and each multiple integral is realized with respect to compensated Poisson random measure ˆη n = η n nµ. Also, since X is a compact set of IR d and ν is a probability measure, one has that, for every n, µ n = nlxνm < and η n is a Poisson random variable of parameter µ n. 4.2 Geometric U-statistics of order 2 Our main result of this section is given in below. Note that items a and b in the forthcoming theorem are a consequence of [8, Theorem 7.3], as well [2]. Theorem 4.. Assume that the kernels h,n and h 2,n are given by the RHS of 4.3. a If h L 2 µ> 0, then as n, one has the exact asymptotic VarF n VarF,n h 2 L 2 µ n3. b Define F n := F n IEF n VarFn, n. Set h,n = VarF,n 2 h,n, and := h,n 3 L 3 µ n = h 3 L 3 µ h 3 L 2 µ n 2. Let N N 0,. Then, there exists a constant C 0,, independent of n, such that d W F n, N C

9 c Denote αh 2,h 2 := 9 h 2 L 2 µ h h 2 2 L 2 µ h 6, and 4.5 L 3 µ ρ h,h 2 := 3 h, h h 2 L 2 µ h L 3 µ If moreover hx, x 2 0, for x, x 2 2 a.e. µ 2, and also α h,h 2 ρ h,h 2 /2, then there exists a constant 0 < c < C such that, for n large enough, c d W F n, N C, and the rate of convergence induced by the sequence := C = Cn /2 is therefore optimal. Before giving a proof of Theorem 4., we present three following lemmas. Lemma 4.. Let h L,2 s 2, µ 2 be a non-negative kernel, and F n be the corresponding geometric U-statistic of order 2 given by 4. based on Poisson random measure η n. Assume that h 0, where the kernels h and h 2 are given by 4.3. Denote F n = F n IEF n VarFn := VarFn {I g,n + I 2 g 2,n }. Then, for any function f F 0, and every z, as n, we have IE 0 f F n + ud z Fn du IEf N. Proof. According to [8, Theorem 7.3], under assumption h 0, we know that the sequence F n converges in distribution to N N 0,. Moreover VarF n n 3, and IEI 2 g 2,n 2 = 2 g 2,n 2 L 2 µ 2 n n2. Fix z. Then by definition of Malliavin derivative, we have D z Fn = VarFn {g,nz + 2I g 2,n z, }. Clearly, using representation of kernels g,n = nh z, and g 2,n = h 2, we obtain that VarF n 2 g,n z = O z n 2 0, as n. For the other term, using the isometry for Poisson multiple integrals, and the representation g 2,n = h 2 in particular g 2,n does not depend on n, we obtain that I g 2,n z, 2 IE = g2 2,n z, xµ ndx VarFn VarF n = O z n 2 0. Hence, D z Fn converges in distribution to zero, and therefore using Slutsky Theorem, for any u 0,, we conclude that the sequence F n + ud z Fn converges in distribution to N N 0,. Now, since f 2, a direct application of dominated convergence theorem completes the proof. Lemma 4.2. Let all notations and assumptions of Lemma 4. prevail. Denote αh 2,h 2 := 9 h 2 L 2 µ h h 2 2 L 2 µ h 6, and 4.7 L 3 µ ρ h,h 2 := 3 h, h h 2 L 2 µ h L 3 µ 9

10 Set g,n = VarF,n 2 g,n, and := g,n 3 L 3 µ. Assume that n α2 h,h 2 0. Then, as n, two dimensional random vector F n, D F n, DL Fn L 2 µ n converges in distribution to two dimensional random vector N, N 2, where N N 0,, N 2 N 0, α 2 h,h 2, and moreover IEN N 2 = ρ h,h 2. Proof. First note that the random variables F n take the form F n = VarFn {I g,n + I 2 g 2,n }, where the kernels g,n and g 2,n are given by 4.3. Recall that under assumption h 0, according to [8, Theorem 7.3], we know that sequence F n converges in distribution to N N 0,. Moreover, representation 4.3 yields that = h 3 L 3 µ h 3 L 2 µ n 2. Now, fix z. Then D z Fn = VarFn {g,nz + 2I g 2,n z, }, and D z L Fn = VarFn {g,nz + I g 2,n z, }. Therefore, D F n, DL { Fn L 2 µ n = g VarF n,nzµ 2 n dz } + 3 g,n zi g 2,n z, µ n dz + 2 Ig 2 2,n z, µ n dz. Using multiplication formula for multiple Poisson integrals, one can deduce that X n : = D F n, DL Fn L 2 µ n { } = 3I g,n g 2,n + 2I g 2,n 2 g 2,n + 2I 2 g 2,n g 2,n. VarF n On the other hand, we know that VarF n n 3 h 2 L 2 µ, as n. Also, the isometry property of Poisson multiple integrals tells us that IEI g,n g 2,n 2 = g,n g 2,n 2 L 2 µ = n n 5 h h 2 2 L 2 µ. Hence IEX2 n n, as n. Let denote Xn := X,n+X 2,n +X 3,n, where And similarly, IEX2,n 2 I 2g 2,n 2 := IE g 2 2,n = On 2 0. VarF n IEX3,n 2 I2 2g 2,n := IE g 2 2,n = On 0. VarF n Hence, in particular these observations imply that sequences X 2,n and X 3,n converge in distribution to 0. Also, 0

11 Moreover, as n : IEX,n 2 I 3g,n : = IE g 2 2,n = 9 h h 2 2 L 2 µ n5 VarF n 2 VarF n 2 as n 9 h 2 L 2 µ h h 2 2 L 2 µ h 6 L 3 µ g,n g 2,n 3 VarF n L 3 µ n 2 0. n = α 2 h,h 2 0. Hence, according to [6, Corollary 3.4], sequence X,n converges in distribution to N 0, σh 2,h 2. Now, [9, Corollary 3.4] also implies that two dimensional random vector I g,n VarFn, I 3g,n g 2,n law N, N 2, VarF n where N N 0,, N 2 N 0, α 2 h,h 2, and moreover I g,n IE VarFn I 3g,n g 2,n = VarF n 3 Var 3 2 F n g,n, g,n g 2,n L 2 µ n n 3 h, h h 2 L 2 µ h 3 L 3 µ = ρ h,h 2 = IEN N 2. Now, an application of multi-dimensional version of Slutsky Theorem completes the proof. Lemma 4.3. Let all notations and assumptions of Lemmas 4. and 4.2 prevail. Assume that ρ h,h 2 2, and N N 0,. Then, for any function f F 0, as n, we have IE f F n F n f F n { } 2 + ρ h,h 2 IEf N. Proof. Recall that according to [8, Theorem 7.3], assumption h 0 implies that d W F n, N 0, Cn /2 for some constant C independent of n. Let f F 0. Then according to Lemma 3., we have IE f F n F n f F n + 2 IE = IE f F n D F n, DL Fn L 2 µ n D z Fn 2 D z L Fn R f nzµ n dz. Note that f. Also, proof of Lemma 4.2 reveals that for some constant M Hence, the sequence sup IE n D Fn, DL 2 Fn L 2 µ n M <. { f F n D F n, DL Fn } L 2 µ n n

12 is uniformly integrable. Therefore, using Lemma 4.2, one can deduce as n that IE f F n D F n, DL Fn L 2 µ n IEf N N 2 = ρ h,h 2 IEf N. Hence, we are left to show that as n, we have 2 IE D z Fn 2 D z L Fn Rnzµ f n dz 2 IEf N. First, note that As a result, D z Fn 2 D z L Fn = VarF n 3 2 { g 3,nz + 5g 2,nz I g 2,n z, + 8g,n z I g 2,n z, 2}. 2 IE D z Fn 2 D z L Fn Rnz f µ n dz = VarF n 3 { 2 IE g 2,nz 3 Rnz f µ n dz + 5IE g,nz 2 I g 2,n z, Rnz f µ n dz + 8IE } g,n z I g 2,n z, 2 Rnz f µ n dz := B,n + B 2,n + B 3,n. Now, using the facts that VarF n VarF,n, and Lemma 4., for the convergence B,n 2 IEf N, it is enough to show that the sequence { VarF n 3 2 g 3 2,nz Rnz f } 2 IEf N µ n dz is uniformly integrable. However, this is clear because of the fact Rnz f 2 IEf N 2, and moreover for any ɛ > 0, we have g,n 3 z +ɛµn dz = On ɛ. n VarF n 3 2 For the term B 2,n, using the fact R f n, and Cauchy Schwarz inequality, we obtain that IE I g 2,n z, Rnz f IEI g 2,n z, 2 = C z n 2 where C z = h 2 z, 2 L 2 µ > 0. Therefore B 2,n = On 2 0, as n. For the last term B 3,n, we again use R f n and isometry for Poisson multiple integrals to obtain that B 3,n VarF n = VarF n g,n z n 3 g2,nz, 2 xµ n dx µ n dz h z h 2 2z, xµdx µdz. 2

13 Now, since VarF n 3 2 n 4, we deduce that in fact B 3,n = On 0, as n. This completes the proof. Proof of Theorem 4.. One just needs to apply Theorem., item III. 4.3 Edge-counting in random geometric graphs In this section, we shall apply our main results to the following situation: η is a compensated Poisson measure on the product space IR +, BIR + where, is a Borel space with control measure given by ν := l µ, 4.9 with ldx = dx equal to the Lebesgue measure and µ equal to a σ-finite Borel measure with no atoms. For every n, we set η n to be the Poisson measure on, given by the mapping A η n A := η[0, n] A A µ, in such a way that η n is a Poisson measure on,, with intensity µ n := n µ. For every n, the random variable F n is a U-statistic of order 2 with respect to the Poisson measure η n := η n + µ n, in the sense of the following definition. Definition 4.2. A random variable F is called a U-statistic of order 2, based on the Poisson random measure η n defined above, if there exists a kernel h L sµ 2 that is, h is symmetric and in L sµ 2, such that F = hx, x 2, 4.0 x,x 2 η 2 n, where the symbol η 2 n, indicates the class of all 2-dimensional vectors x, x 2 such that x i is in the support of η n i =, 2 and x x 2. We recall that, according to the general results proved in [2, Lemma 3.5 and Theorem 3.6], one has that, if a random variable F as in 4.0 is square-integrable, then necessarily h L 2 sµ 2, and F admits a representation of the form F = IEF + F + F 2 := IEF + I h + I 2 h 2, 4. where I and I 2 indicate multiple Wiener-Itô integrals of order and 2, respectively, with respect to η, and h t, z := 2 [0,n] t ha, z µ n da 4.2 = 2 [0,n] t [0,n] sha, z νds, da = 2 [0,n] tn IR + ha, z µda L 2 µ, h 2 t, x, t 2, x 2 := [0,n] 2t, t 2 hx, x 2, 4.3 where ν is defined in 4.9. Let the framework and notation of Section 4. prevail, set = IR d, and assume that µ is a probability measure that is absolutely continuous with respect to the Lebesgue measure, with a density f that is bounded and everywhere continuous. It is a standard result that, in this case, the non-compensated Poisson measure η n has the same distribution as the point process N n A δ Yi A, A BIR d, i= 3

14 where δ y indicates the Dirac mass at y, {Y i : i } is a sequence of i.i.d. random variables with distribution µ, and N n is an independent Poisson random variable with mean n. Throughout this section, we consider a sequence {t n : n } of strictly positive numbers decreasing to zero, and consider the sequence of kernels {h n : n } given by h n : IR d IR d IR + : x, x 2 h n x, x 2 := 2 {0< x x 2 t n}, where, here and for the rest of the section, stands for the Euclidean norm in IR d. Then, it is easily seen that, for every n, the U-statistic F n := h n x, x 2, 4.4 x,x 2 η 2 n, equals the number of edges in the random geometric graph V n, E n where the set of vertices V n is given by the points in the support of η n, and {x, y} E n if and only if 0 < x y t n in particular, no loops are allowed. We will now state and prove the main achievement of the section, refining several limit theorems for edge-counting one can find in the literature see e.g. [2, 7, 8, 20, 2, 22] and the references therein. Observe that, quite remarkably, the conclusion of the forthcoming Theorem 4.2 is independent of the specific form of the density f.for every d, we denote by κ d the volume of the ball with unit radius in IR d. Theorem 4.2. Assume that nt d n, as n. a As n, one has the exact asymptotics IEF n κ d n 2 t d n 2 IRd fx 2 dx, VarF n κ2 d n3 t d n 2 4 IR d fx 3 dx. 4.5 b Define F n := F n IEF n VarFn, n, and let N N 0,. Then, there exists a constant C 0,, independent of n, such that d W F n, N := Cn / c If moreover n t d n 3, then there exists a constant 0 < c < C such that, for n large enough, c n /2 d W F n, N C n /2, and the rate of convergence induced by the sequence = C n /2 is therefore optimal. Proof. The two asymptotic results at Point a follow from [20, Proposition 3.] and [20, formula 3.23], respectively. Point b is a special case of the general estimates proved in [8, Theorem 3.3]. In order to prove Point c it is therefore sufficient to show that the sequence F n verifies Assumptions ii, iii and iv of Theorem.-II, with respect to the control measure ν defined in 4.9, and with values of α, β and ρ such that α ρ β/2. First of all, in view of 4., one has that, a.e. dt µdz, D t,z Fn = VarFn {h,nt, z + 2I h 2,n t, z, }, where the kernels h,n and h 2,n are obtained from 4.3 and 4.3, by taking h = h n. Since h,n t, z = 2 [0,n] tn IR d hz, aµda, we obtain that VarF n 2 h,n t, z = 4

15 Ot 2d n n 2 0, as n. Also, using the isometric properties of Poisson multiple integrals, I h 2,n t, z, 2 n IE IR d h nz, xµdx VarFn VarF n = Ont d n 2 0. It follows that D t,z Fn converges in probability to zero for dν-almost every t, z IR + IR d, and Assumption ii of Theorem.-II is therefore verified. In order to show that Assumption iii in Theorem.-II also holds, we need to introduce three standard auxiliary kernels: h 2,n 2 h 2,n t, x := [0,n] tn h 2 nx, aµda IR d h 2,n h 2,n t, x, s, y := [0,n] t [0,n] sn h n x, ah n y, aµda IR d h,n h 2,n t, x := 2 [0,n] tn 2 h n a, yh n x, yµdaµdx. IR d IR d The following asymptotic relations for n can be routinely deduced from the calculations contained in [8, Proof of Theorem 3.3] recall that the symbol indicates an exact asymptotic relation, and observe moreover that the constant C is the same appearing in 4.6: h 2,n 2 h 2,n 2 L 2 ν = On 3 t d n h 2,n h 2,n 2 L 2 ν 2 = On 4 t d n h,n h 2,n 2 L 2 ν κ4 d n5 t d n 4 fx 5 dx IR d h,n, h,n h 2,n L 2 ν κ3 d n4 t d n 3 fx 4 dx IR d VarF n C κ2 d n5/2 t d n 2 fx 3 dx IR d h,n h 2,n 3 L 3 ν = On 7 t d n h,n 3 L 3 ν κ 3 d n4 t d n 3 IR d fx 4 dx 4.23 h,n 4 L 4 ν = On 5 t d n h 2,n 2 L 2 ν = On 2 t d n Using the fact that, by definition, L Y = q Y for every random variable Y living in the qth Wiener chaos of η, we deduce that using the control measure ν defined in 4.9 D F n, DL Fn L 2 ν = { VarF n + 3 h,n t, zi h 2,n t, z, νdt, dz + 2 IR + IR d h 2,nt, zνdt, dz IR + IR d } Ih 2 2,n t, z, νdt, dz. IR + IR d Using a standard multiplication formula for multiple Poisson integrals see e.g. [7, Section 6.5] on the of the previous equation, one deduces that D F n, DL Fn L 2 ν = { } 3I h,n h 2,n + 2I h 2,n 2 h 2,n + 2I 2 h 2,n h 2,n VarF n := X,n + X 2,n + X 3,n. 5

16 Now, in view of 4.7, 4.8 and 4.2, one has that, as n, Also, 4.9 yields that IEX 2 2,n = Ont d n 2 0, and IEX 2 3,n = Ont d n 0. IEX 2,n 9 IR d fx5 dx C 2 IR d fx3 dx 2 := α2 > 0. Finally, in view of 4.22 and 4.23, h,n h 2,n 3 VarF n, h,n 3 L 3 ν VarF n = L 3 ν On 2 0, and A standard application of [9, Corollary 3.4] now implies that, as n I h,n VarFn, I 3h,n h 2,n law, 2, 4.26 VarF n where N 0, and 2 N 0, α 2 are two jointly Gaussian random variables such that ρ 3 := IE 2 = lim n Var 3 2 F n h,n, h,n h 2,n L 2 ν = 2 IR d fx4 dx C IR d fx3 dx 3/2. Now, since relation 4.25 implies that VarF n /2 I 2 h 2,n converges to zero in probability, we deduce that the sequence F n, D F n, DL Fn L 2 ν, n, converges necessarily to the same limit as the one appearing on the RHS of We therefore conclude that Assumption iii in Theorem.-II is verified with α defined as above, and ρ := ρ /α. To conclude the proof, we will now show that Assumption iv in Theorem.-II is satisfied for u n = h 3,n VarF n 3/2 and β = 8 IR d fx4 dx C IR d fx3 dx 3/ To see this, we use again a product formula for multiple stochastic integrals to infer that IE D t,z Fn 2 D t,z L Fn Rnt, f z νdt, dz IR + IR d = VarF n 3 { 2 IE h 3,nt, z Rnt, f z νdt, dz IR + IR d + 5IE h 2,nt, z I h 2,n t, z, Rnt, f z νdt, dz IR + IR d } + 8IE h,n t, z I h 2,n t, z, 2 Rnt, f z νdt, dz IR + IR d := B,n + B 2,n + B 3,n. Since relations 4.23 and 4.24 imply that, as n and using the notation 4.27, u n t, z IR + IR d νdt, dz β, and IR + IR d 6 un t, z 4/3 νdt, dz = On /3,

17 the proof is concluded once we show that IEB 2,n, IEB 3,n 0. In order to deal with B 2,n, we use the fact that Rn f, together with the Cauchy-Schwarz and Jensen inequalities and the isometric properties of multiple integrals, to deduce that IR IE h 2,nt, z I h 2,n t, z, R f nt, z νdt, dz + IR d 2 n 7/2 h n z, aµda h 2 nz, aµda µdz IR d IR d IR d 2 n 7/2 h n z, aµda h IR IR 2 nz, aµda µdz d d IR d = On 7/2 t d n 3/2. Since we work under the assumption that nt d n 3, this yields that, as n, IEB 2,n = On /2 t d n 3/2 0. Analogous computations yield that IEB 3,n = On t d n 0, and this concludes the proof. 5 Connection with generalized Edgeworth expansions In this section, we briefly explain the link with generalized Edgeworth expansion. For more details, we refer the reader to [3], [2, Chapter 5] and [4, Appendix A]. Let X and X 2 be two real valued random variables, with finite moments up to some order M {, 2, } {+ }. For i =, 2, write {κ p X i : p =,, M} for the associated sequence of cumulants. Define the formal cumulants associated with X, X 2 as κ f px, X 2 := κ p X κ p X 2, p =,, M. The formal moments associated with the sequence {κ p X, X 2 }, denoted by {m f px, X 2 } are recursively defined as m f 0 X, X 2 = 0 and for p =,, M: p p m f px, X 2 := κ f q+ q X, X 2 m f p q X, X 2. q=0 Example 5.. In general, we have that m f X, X 2 = κ f X, X 2, 5. m f 2 X, X 2 = κ f X, X κ f 2 X, X 2, 5.2 m f 3 X, X 2 = κ f X, X κ f X, X 2 κ f 2 X, X 2 + κ f 3 X, X In particular, if κ p X = κ p X 2 for p =, 2, then m f px, X 2 = 0 for p =, 2. Moreover, if κ 3 X 2 = 0, then m f 3 X, X 2 = κ 3 X. Definition 5.. Let h be a Mth differentiable function with bounded derivatives up to order M. We define the generalized Mth Edgeworth expansion E M X, X 2, h of IEhX around IEhX 2 by E M X, X 2, h := IEhX 2 + M p= m f px, X 2 IEh p X p! 7

18 In particular, if X 2 = N N 0,, and X is a random variable such that κ X = 0 and κ 2 X =, then we obtain the generalized third-order Edgeworth expansion of IEhX around IEhN as E 3 X, N, h = IEhN + κ 3X IEh 3 N 3! = IEhN + κ 3X IEhN H 3 N, 3! where H p stands for Hermite polynomial of order p defined as: H p x = p e x2 2 d p 2 dx p e x 2. For example H 0 x =, H x = x, H 2 x = x 2 and H 3 x = x 3 3x. We also need two following properties of Hermite polynomials: i H p = ph p and ii xh p x = H p+ x + ph p x. 5.5 We continue with the following general lemma. Lemma 5.. Let N N 0,. Assume that h Cb, i.e continuously differentiable with bounded derivative. Denote by f h the corresponding solution to the Stein s equation 2.7 associated to h. Then the function f h is twice differentiable, and moreover for any p 2 IE hn H p+ N = p + IE H p 2 Nf h N. In particular, for p = 2, we obtain IEhN H 3 N = 3 IEf h N. Proof. The fact that the solution f h is twice differentiable with bounded derivatives is indeed Lemma 2.3, item ii. Now, since f h is the solution of Stein s equation 2.7, we have f h u f hu u f h u = h u. 5.6 Multiply both sides of 5.6 by second Hermite polynomial H p, and compute mathematical expectation evaluated at N N 0,, and using property ii in 5.5, we obtain IEH p Nf h N IEH pnf h N IEH p+ Nf h N pieh p Nf h N = IEH pnh N. Now, using property i in 5.5, and integration by parts formula [4, Theorem 2.9.], we infer that in fact Now the claim follows immediately. IEH p+ Nf h N = IEH pnf h N, IEH p Nf h N = IEH p 2 f h N, and IEH p Nh N = IEH p+ NhN. Lemma 5.2. Let all notations and assumptions of Lemmas 4. and 4.2 prevail. As before, we denote X n := D F n, DL Fn L 2 µ n. Then, as n, the following asymptotic relations take place: IE F n X n 3 h,h h 2 L 2 µ h 3 L 2 µ n 2, 2 κ 3 F n h,h 0 h L 2 µ +6 h 2,h 0 0 h L 2 µ 2 h 3 L 2 µ 8 n 2,

19 3 κ 4 F n C h,h 2 n, for some constant C h,h 2 depending only on the norms and inner products of kernels h and h 2. Proof. The claims are direct consequences of straightforward computations based on multiplication formula for multiple Poisson integrals together with the isometry property. Remark 5.. Unlike Wiener structure see [4, Proposition 9.4.], for Poisson multiple integrals seems it is impossible to express the conditions in Lemma 4.2 in terms of cumulants κ 3 F n and κ 4 F n. However in virtue of Lemma 5.2, one can easily deduce that for some constant C h,h 2 depending only on kernels h and h 2 that, as n : IE F n D F n, DL Fn L 2 µ n κ 3 C F n h,h 2 κ 4 F. n Theorem 5.. Let all notations and assumptions of Lemmas 4. and 4.2 prevail. Then, as n, Moreover, for any function g Cb 3, we have, as n κ 3 F n 2 2 ρ h,h IEg F n E 3 F n, N, g IEf g N. Proof. Using the fact that = h 3 L 3 µ h 3 L 2 µ n 2, Lemma 5.2, item ii, and h, h 0 0 h L 2 µ = h 3 L 3 µ, we infer that, as n : κ 3 F n + 6 h 2, h 0 0 h L 2 µ 2 h 3. L 3 µ Now, taking into account that h 2, h 0 0 h L 2 µ 2 = h, h h 2 L 2 µ, the convergence implication 5.7 follows at once. For the second part, using Lemma 5., we infer that IEg F n E 3 F n, N, g = IEg F n IEgN = IEf g F n F n f g F n := A,n + A 2,n. κ 3 F n 3! IEgNH 3N + κ 3 F n IEf g N 2 Now, according to Lemma 4.3, we have A,n { 2 +ρ h,h 2 }IEf g N, and claim follows. Remark 5.2. It is worth to mention that in Wiener structure, it is well known that see [4, Proposition 9.3.], replacing IEgF n IEgN with IEgF n E 3 F n, N, g actually increases the rate of convergence to zero. However, Theorem 5. reveals that this is not the case for sequence F n of geometric U-statistcs of order two. A decisive reason to explain this phenomenon is existence of an extra term in fact the second term in RHS of 2.8 in the upper bound for the Wasserstein distance for normal approximation of Poisson functionals compare to Wiener functionals. 9

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