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U Gamma limits and -statistics on the Poisson space Giovanni Peccati∗ and Christoph Thäle†‡ Abstract UsingStein’smethodandtheMalliavincalculusofvariations,wederiveexplicitestimatesforthe Gammaapproximationoffunctionals ofa Poissonmeasure. Inparticular,conditionsarepresented under which the distribution of a sequence of multiple Wiener-Itoˆ stochastic integrals with respect to a compensated Poisson measure converges to a Gamma distribution. As an illustration, we 3 present a quantitative version and a non-central extension of a classical theorem by de Jong in 1 0 the case of degenerateU-statistics of order two. Severalmultidimensional extensions,in particular 2 allowing for mixed or hybrid limit theorems, are also provided. n a Keywords. Chaos; Contraction; Gamma distribution; De Jong’s theorem; Malliavin calculus; J Mixed limit theorem; Multiple stochastic integral; Non-central limit theorem; Poisson process; 0 Stein’s method; U-statistic. 3 MSC2010. Primary 60F05, 60G55; Secondary 60H05, 60H07, 62E20. ] R 1 Introduction P . h The use of the Malliavin calculus of variations in order to deduce limit theorems for non-linear func- t a tionals of random measures has recently become a relevant direction of research, one reason for that m being the many successful applications in geometric probability or stochastic geometry. Apart from [ a few exceptions, most contributions to this topic fall into the two categories of normal and Poisson 1 approximations; see [3,10,11,13,22,24, 30, 38]for distinguishedexamples of theformerclass, mostly v based on theuseof the Stein’s method(cf. [18]); see[2,21, 34] for references based on thecombination 9 8 of Malliavin calculus and of the Chen-Stein method for Poisson approximations. We also refer to [7] 2 for recent extensions to general absolutely continuous distributions having support equal to the real 7 . line. 1 0 The aim of the present paper is to provide the first array of results concerning limit theorems on 3 the Poisson space, where the limit distribution is absolutely continuous and has support contained 1 in a proper subset of R. More precisely, we are interested in probabilistic approximations where the : v limiting random variable has a centred Gamma distribution Γ with parameter ν > 0. We say that a i ν X d random variable G(ν) has distribution Γ if G(ν) = 2F(ν/2) ν, where F(ν/2) has a usual Gamma ν r − d a distribution with mean and variance both equal to ν/2 (here and throughout = stands for equality in distribution). If ν 1 is an integer, then Γ reduces to the centred χ2-distribution with ν degrees of ν ≥ freedom. We remark that the support of Γ is given by the half-line [ ν,+ ), and that the first four ν − ∞ moments of Γ are 0, 2ν, 8ν and 12ν2+48ν, respectively. We will often meet these expressions in the ν discussion to follow. Our main contribution is the general estimate stated in Theorem 2.1, which involves Malliavin operators and is obtained by means of Stein’s method, allowing one to measure the distance between the law of a given Poisson functional and Γ . This estimate is applied to deduce explicit sufficient ν ∗Luxembourg University, Mathematics Research Unit, Campus Kirchberg, G 221, L-1359 Luxembourg. E-mail: [email protected] †Ruhr-University Bochum, Faculty of Mathematics, NA 3/68, D-44781 Bochum, Germany. E-mail: chris- [email protected] ‡The second author has been supported by the German Research Foundation (DFG) via SFB/TR-12 “Symmetries and Universality in Mesoscopic Systems”. 1 conditions for Gamma limit theorems involving sequences of multiple Wiener-Itô stochastic integrals. Our analysis is significantly inspired by [15, 16], where the problem addressed in the present paper was first studied in the framework of non-linear functionals of general Gaussian fields. However, due to the combinatorial complications one has to face when dealing with point measures, our paper contains a number of new subtle computations related to the explicit estimation of Malliavin operators on configuration spaces. One specific problem we will have to deal with is that the solution of the Stein’s equation associated with the law of G(ν) is not differentiable at x = ν. Thus, in order to − obtain bounds that are well-suited for our applications (which may involve random variables possibly taking values in ( , ν)), we will have to combine techniques recently introduced by Schulte [33] −∞ − with classical isometric formulae borrowed from the standard reference [27]; see Proposition 2.3 below. One should note that, in view of the exact chain rules that are available on a Gaussian space, the non-differentiability of the Stein solution in one point is immaterial when studying the Gamma approximation of smooth functionals of a Gaussian field; see again [15, 16]. As an illustration, we will include some applications to non-central limit theorems for sequences of degenerate (in the sense of Hoeffding) U-statistics. Our findings generalize several classic result in the field; cf. [1, 8]. In particular, we derive a quantitative and a non-central version of a famous theorem by P. de Jong [4, 5]. Our analysis also contains a quantitative version of a non-central result recently discussed by Reitzner and Schulte [30, Section 5.1]. Finally, to demonstrate the flexibility and scope of our approach, we will show that our analysis cannaturallybeextendedtoamultidimensionalframework. Wewillnotonlyobtainmultidimensional Gammalimittheorems, butalsomixedorhybridresults,wherethemultidimensionallimitdistribution iscomposedbothofGammaandofnormalorPoissoncomponents. Thiskindoflimittheoremsheavily relies on our use of Malliavin operators. We are not aware of any other available technique allowing one to deduce general mixed limit results, such as the ones deduced in the present paper. We shall see that ourfindingsarearefinementofthe‘Portmanteau inequalities’ recently obtained byBourguin and Peccati in [2]. In this respect, we stress that our results will implicitly yield a collection of sufficient conditions in order to have that two sequences of Poisson functionals are asymptotically independent. This provides a new contribution to the difficult and mostly open problem of characterizing the asymptotic and non-asymptotic independence of functionals of a Poisson measure; see e.g. [28, 31]. The remainder of the paper is organized as follows. In Section 2 we present our results in full generality. Some background material is collected in Section 3, whereas the final Section 4 contains detailed proofs, as well as some ancillary technical results. 2 Presentation of the results We will now present an overview of the main findings of the paper. To enhance the readability of our text, we have gathered together in Section 3 definitions, notation and relevant results from the literature. 2.1 General limit theorems Every random object considered below is defined on a suitable probability space (Ω, ,P). The F approximation results obtained in the present paper deal with (real-valued) functionals of a Poisson measureη onsomePolish space( ,Z)havingnon-atomic andσ-finitecontrolµ; seeSection 3-(I).We Z will assume that these functionals are square-integrable random variables. To measure the distance between the distribution of a functional F of η and that of a centred Gamma random variable G(ν), we shall use the (pseudo-) metric d , which is defined as follows: for every pair of square-integrable 3 random variables X,Y, we put d (X,Y) = sup E[h(X)] E[h(Y)] , 3 − h∈H3 (cid:12) (cid:12) (cid:12) (cid:12) 2 where 3 := h 3 : h(j) 1, j 1,2,3 (with h(j) the derivative of order j of h), and where ∞ H { ∈ C k k ≤ ∈ { }} 3 is the space of thrice differentiable functions on R having bounded derivatives. We notice that the C topology induced by d is stronger than the topology induced by convergence in distribution, which 3 implies thatif d F ,G(ν) 0, as n , forsomesequenceof functionals F , thenthedistribution 3 n n → → ∞ of F converges to Γ . By a slight abuse of notation, and to stress the role of the underlying Gamma n ν (cid:0) (cid:1) distribution, we shall often write d (F,Γ ) instead of d F,G(ν) . 3 ν 3 Forq 1,wewriteL2(µq)toindicatetheHilbertspaceofBorel-measurablefunctionalson q that (cid:0) (cid:1) ≥ Z are square-integrable with respect to µq. We also use the following special notation: L2(µ1)= L2(µ), and L2 (µq) is the subspace of L2(µq) composed of those functions that are µq-a.e. symmetric; see sym Section 3-(II). Moreover, in order to simplify the notation, we use the convention that and , k · k h· ·i stand for the norm and the scalar product in some space L2(µq) whose order q will always be clear from the context. Our first result is a quantitative estimate for d F,Γ in terms of the Malliavin operators D and 3 ν L−1, that is, the derivative operator and the pseudo-inverse of the Ornstein-Uhlenbeck generator. We (cid:0) (cid:1) recall that the derivatives DF and DL−1F are random elements with values in the Hilbert space L2(µ); see Section 3-(V). Theorem 2.1 (General Gamma bounds). Let F be a centred and square-integrable functional of the Poisson measure η, and assume that F is in the domain of the derivative operator D. Then, (2.1) d (F,Γ ) c A (F)+c A +2c A (F) 3 ν 1 1 2 2 1 3 ≤ := c E 2(F +ν) DF, DL−1F +c E[D F 2 D L−1F ]µ(dz) 1 + 2 z z −h − i | | | | ZZ (cid:12) (cid:12) (cid:12) +2c E (D 1(cid:12) )(D F)D L−1F µ(dz), 1 z {F>−ν} z z | | ZZ (cid:2) (cid:3) with constants c and c given by 1 2 c =max(1,1/ν +2/ν2) and c = max(2/3,2/(3ν) 3/ν2 +4/ν3). 1 2 − If in addition E DF, DL−1F F 0 (a.s.-P), then h − i| ≥ (cid:2) (cid:3) A (F) A′(F) := E (2(F +ν) DF, DL−1F )2 , 1 ≤ 1 −h − i q and consequently (cid:2) (cid:3) (2.2) d (F,Γ ) c A′(F)+c A +2c A (F) 3 ν ≤ 1 1 2 2 1 3 Remark 2.2. (i) In (2.1), we implicitly used a ‘trajectorial’ definition of the random function z D 1 , that is, we put D 1 = 1 1 , without necessarily 7→ z {F>−ν} z {F>−ν} {F+DzF>−ν} − {F>−ν} assuming that E (D 1 )2µ(dz) < z {F>−ν} ∞ ZZ (note that this last relation is equivalent to the fact that 1 belongs to the set domD, as {F>−ν} defined in Section 3-(V); see Lemma 3.1). It is easily checked that (D 1 )(D F) =(1 +1 )D F , z {F>−ν} z {F≤−ν<F+DzF} {F+DzF≤−ν<F} | z | in such a way that A (F) 0. An effective bound on A (F), in the case where µ is a finite 3 3 ≥ measure and F is a multiple Wiener-Itô integral, is presented in Proposition 2.3. (ii) As first done in [22], we shall often control the quantity A (F) appearing in (2.1) by using the 2 relation 1/2 1/2 (2.3) A (F) A (F) A (F) := E[D F 4]µ(dz) E[D L−1F 2]µ(dz) . 2 4 5 z z ≤ × | | × | | (cid:18)ZZ (cid:19) (cid:18)ZZ (cid:19) 3 We also note that, if F : n 1 is a sequence of random variables with bounded variances n { ≥ } living in a fixed sum of Wiener chaoses, then the numerical sequence n A (F ) is necessarily 5 n 7→ bounded. (iii) Theorem 2.1 should be compared with the following bound from [16, Theorem 3.11]. Let F be a centered functional of a Gaussian measure on with control µ, and assume that F is in the Z domain of the Malliavin derivative D (see [18, Chapter 2] for relevant definitions), then there exists a constant K such that, for some adequate distance d, d(F,Γ ) K E 2(F +ν) DF, DL−1F . ν + ≤ × −h − i (cid:12) (cid:12) The presence of the additional term (cid:12) (cid:12) c E[D F 2 D L−1F ]µ(dz)+ 2c E (D 1 )(D F)D L−1F µ(dz) 2 z z 1 z {F>−ν} z z | | | | | | ZZ ZZ (cid:2) (cid:3) in (2.1) or (2.2) is due to the characterization of the Malliavin derivative on the Poisson space as a difference operator as well as to the non-differentiability at ν of the solution of the − Stein-equation characterizing Γ ; see Section 3-(V). As proved in [16, Proposition 3.9], on the ν Gaussian-Wiener space the condition E DF, DL−1F F 0 (a.s.-P) is satisfied for every F h − i| ≥ in the domain of D. (cid:2) (cid:3) (iv) Other relevant one-dimensional bounds for probabilistic approximations involving Malliavin op- eratorsonthePoissonspaceareprovedin[22],dealingwithnormalapproximations, [21],dealing with the Poisson approximation of integer-valued random variables and [7], focusing on absolutely continuous distributions whose support is given by the real line. See [2, 24] for several multidimensional extensions. As announced, we conclude the present section with a useful bound on the quantity A (F), in the 3 case where F = I (f) equals a multiple Wiener-Itô integral and the control measure µ is finite. At the q cost of a heavier notation, our techniques could suitably be modified in order to deal with the case of a random variable F having a finite chaotic expansion. Proposition 2.3. Let the control measure µ be finite, and consider F = I (f), where q 2 and q ≥ f L2 (µq). We assume that (i) E (D F)4µ(dz) < , that (ii) the random function ∈ sym Z z ∞ R z D F D F := v(z) z z Z ∋ 7→ | | is such that v(z) domD for µ(dz)-almost every z, and satisfies ∈ E (D v(z ))2µ(dz )µ(dz ) < . z2 1 1 2 ∞ ZZZZ Then, defining A (F) as in (2.1), one has the bound 3 q A (F) E (D F)4µ(dz)+ E (D D F)2(D F)2µ(dz )µ(dz ) (2.4) 2√2 3 ≤s ZZ z s ZZZZ z2 z1 z1 1 2 + E (D D F)4µ(dz )µ(dz ). z2 z1 1 2 s ZZ ZZ Remark 2.4. (i) Another way of controlling the term A (F), whenever F has a finite chaotic ex- 3 pansion, is discussed in [33]. Oneshouldnote that, albeit ourproof of Proposition 2.3 also starts with an integration by parts formula, our strategy for controlling the term A (F) is significantly 3 different. Indeed, our approach is based on isometric formulae for divergence operators, whereas [33] uses a direct estimation consisting in controlling DF by a random function having a finite | | chaotic expansion. When applied to our framework in the case q > 2, the technique used in 4 [33] leads to expressions involving contractions of the absolute value of the kernel f, therefore producing bounds that are systematically larger than ours. When applied to the case q = 2, the strategy adopted in [33] leads to slower rates of convergence, but allows in principle to dispense with the assumption that the underlying cont rol measure has finite mass. Since all our applications concern sequences of control measures having a finite mass, and for the sake of conciseness, we will omit a formal discussion of this fact. (ii) From the standpoint of geometric applications, focusing on Poisson measures having a finite control is barely a restriction. Indeed, the kind of geometric limit theorems we are interested in typically involve either functionals of a Poisson measure having a finite control, whose total mass asymptotically explodes (like the ones we consider in the applications developed later in thepaper), or functionals of therestriction of a Poisson measure toa finitewindow withgrowing volume; see e.g. [2, 3, 10, 11, 13, 21, 30, 34] for a recent collection of distinguished examples. 2.2 Simplified estimates for supports contained in a half-line The applications we are interested in require that the we consider random variables possibly taking valuesinthehalf-line( , ν),insuchawaythattheratherunusualtermA (F)cannotbedispensed 3 −∞ − with. However, if oneis only interested in measuringthe distance between Γ and thelaw of a random ν variable with support in [ ν,+ ), then the statement of Theorem 2.1 can be significantly simplified, − ∞ since in this case the term A (F) disappears. In particular, whenever the law of F satisfies these 3 requirements, the finiteness of the measure µ does not play any role. This point is made clear in the next statement whose easy proof is left to the reader. Proposition 2.5. Let F be a centered square-integrable functional of the random measure η. Assume that the law of F has support in [ ν,+ ) and that F is in the domain of the derivative operator D. − ∞ Then, the bound (2.1) holds with A (F) = 0. If moreover E DF, DL−1F F 0 (a.s.-P), then the 3 h − i| ≥ estimate (2.2) holds with A (F) = 0. 3 (cid:2) (cid:3) 2.3 General results for sequences of multiple integrals We now focus on the following setup. Let ( ,Z) be a fixed Polish space as above, and η : n 1 n Z { ≥ } be a sequence of Poisson random measures on ( ,Z), such that, for each n, the non-atomic control Z measure µ of η is finite. In view of applications, we allow that µ ( ) , as n . For a given n n n Z → ∞ → ∞ even integer q 2, we consider a sequence I (f ) :n 1 of multiple Wiener-Itô stochastic integrals q n with the follow≥ing characteristics: (a) f{: n 1 ≥ }L2 (µq) is composed of kernels satisfying { n ≥ } ⊂ sym n the technical assumptions stated in Section 3-(VIII) below, and (b) for every n 1, the integral ≥ I (f ) is realized with respect to the compensated Poisson measure ηˆ = η µ . The next theorem q n n n n − characterizes the convergence of the distribution of I (f ), as n , to the limit law Γ . The set q n ν → ∞ of analytic conditions appearing below is expressed in terms of (possibly symmetrized) contraction kernels, whose definition is provided in Section 3-(VI). Observe in particular that f ⋆0f = f2. n q n n Theorem 2.6 (Gamma limits in the Poisson-Wiener chaos). Let the above assumptions and notation prevail (in particular, µ is a finite measure for every n), let q 2 be an even integer and n let f : n 1 L2 (µq) be such that lim q! f 2 = 2ν, and suppose t≥hat the technical conditions { n ≥ } ⊂ sym n n→∞ k nk of Section 3-(VIII) are satisfied. Assume in addition that 4 (2.5) lim f ⋆ℓ f = 0 and lim f ⋆q/2f c f = 0 with c = n→∞k n r nk n→∞k n q/2 n− q nk q q ! q 2 2 q/2 for all pairs (r,ℓ) such that either r = q and ℓ e= 0, or r 1,...,q , ℓ 1,...(cid:0),m(cid:1)i(cid:0)n(r,(cid:1)q 1) and ∈ { } ∈ { − } r and ℓ are not equal to q/2 at the same time. Then, the distribution of I (f ) converges to Γ as q n ν n . Moreover, for some positive finite constant K independent of n, → ∞ d (I (f ),Γ ) c A (I (f ))+c A (I (f )) A (I (f ))+2c A (I (f )) 3 q n ν 1 1 q n 2 4 q n 5 q n 1 3 q n ≤ × (2.6) K max q! f 2 2ν ; f ⋆p f ; f ⋆ℓ f 1/2; f ⋆q/2f c f 0, ≤ × k nk − k n p nk k n r nk k n q/2 n− q nk → (cid:8)(cid:12) (cid:12) (cid:9) (cid:12) (cid:12) 5 e where we have used the notation introduced in (2.1)–(2.3), and the maximum is taken over all p = 1,...,q 1 such that p = q/2 and all (r,ℓ) such that r = ℓ and either r = q and ℓ = 0, or r 1,...,q − 6 6 ∈ { } and ℓ 1,...,min(r,q 1) . ∈ { − } Example 2.7. (i) Assume q = 2. Then, c = 1 and the maximum in (2.6) is taken over the 2 following four quantities: 2 f 2 2ν , f ⋆0f 1/2, f ⋆1f 1/2, f ⋆1f f . k nk − k n 2 nk k n 2 nk k n 1 n− nk (cid:12) (cid:12) (ii) Assume q = 4. (cid:12)Then, c =(cid:12)1/18 and the maximum in (2.6) is teaken over the following ten 4 quantities: 2 f 2 2ν , f ⋆1f , f ⋆0f 1/2, f ⋆1f 1/2, f ⋆2f 1/2, f ⋆3f 1/2, k nk − k n 1 nk k n 4 nk k n 4 nk k n 4 nk k n 4 nk k(cid:12)(cid:12)fn⋆13fnk1/2(cid:12)(cid:12), kfn⋆23fnk1/2, kfn⋆12fnk1/2, kfn⋆11fn−18−1fnk, where we have used the fact that f ⋆1f = f ⋆3f . k n 1 nk k n 3 nek Remark 2.8. (i) Under the assumptions in the statement, one has that the sequence 1/2 A (I (f )) := E[D L−1I (f )2]µ (dz) 5 q n z q n n | | (cid:18)ZZ (cid:19) is such that 2ν A (I (f ))2 = (q 1)! f 2 > 0 as n . 5 q n n − k k → q → ∞ It follows that our inequality (2.6) not only provides an analytic bound in the distance d , but 3 also ensures that the three numerical sequences A (I (f )) : n 1 , A (I (f )) : n 1 and 1 q n 3 q n { ≥ } { ≥ } A (I (f )) : n 1 (all related to Malliavin operators) converge to zero. This fact is crucial 4 q n { ≥ } when dealing with the multidimensional results discussed in Section 2.6. An analogous remark applies to Proposition 2.9 and Theorem 2.13 below. (ii) Similar conditions (only involving contractions of the type ⋆r, with r = 1,...,q 1) in the case r − of multiple integrals with respect to a Gaussian measure can be found in [15, Theorem 1.2]. Non-central results of a similar flavor, in the context of free probability and multiple integrals with respect to a free Brownian motion, are proved in [17]. (iii) We were able to deduce meaningful conditions for Gamma approximations only in the case of an even integer q 2. However, unlike in the Gaussian case (see [15, Remark 1.3]), in a Poisson ≥ framework one cannot exclude a priori the existence of a sequence of multiple integrals of odd orderconvergingtoalimitingGammadistribution. Weprefertoconsiderthisissueasaseparate problem, and keep it as an open direction for future research. (iv) In the estimate (2.6), and in contrast to the main bounds on normal approximations proved in [22], norms of the type f ⋆ℓ f , r = ℓ, appear under a square root. This phenomenon seems k n r nk 6 unavoidable, and it is directly related to the presence of cross terms arising from the specific form of the Stein equation associated with the Gamma distribution. Thefollowing statement shows thatcondition (2.5)mighttake aparticularly attractive form inthe case of double Poisson integrals. This will be used in order to prove the results presented in Section 2.4, dealing with the Gamma approximation of degenerate U-statistics. Proposition 2.9 (Three moments suffice for Gamma approximations). Let the control measures µ : n 1 be finite, let q = 2 and let f : n 1 L2 (µ2) be such that lim E[I2(f )] = { n ≥ } { n ≥ } ⊂ sym n n→∞ 2 n lim 2 f 2 = 2ν, and such that the technical conditions of Section 3-(VIII) are satisfied. Assume in n n→∞ k k 6 addition that f4dµ2 0 and that E[I4(f )] < for every n. Then, condition (2.5) is verified if Z n n → 2 n ∞ and only if R (2.7) E[I4(f )] 12E[I3(f )] 12ν2 48ν as n . 2 n − 2 n −→ − → ∞ In particular, if the sequence F4 is uniformly integrable, then (2.5) and (2.7) are both necessary and n sufficient in order to have that the distribution of F converges to Γ in the sense of the distance d . n ν 3 2.4 An extension of de Jong’s theorem for degenerate U-statistics In the present and the subsequent section, we shall work within the following framework. We fix an integer d 1, and let Y = Y : i 1 be a sequence composed of i.i.d. random variables with values i ≥ { ≥ } in Rd, whose common distribution has a density p(x) with respect to the Lebesgue measure on Rd (written dx). The sequence N(n) : n 1 of integer-valued random variables is independent of Y { ≥ } and such that, for every n, N(n) has a Poisson distribution with parameter n. It is well-known that, in this setting, the random point measure N(n) (2.8) η := δ n Yi i=1 X (where δ represents the Dirac mass at y) is a Poisson measure on = Rd (equipped with the y Z standard Borel σ-field B(R)⊗d) with control measure µ (dx) = np(x)dx. We shall also use the n shorthand notation µ(dx):= µ (dx)= p(x)dx. 1 Ouraim belowistoprovideaGamma-typecounterparttoafamoustheorembyP.deJong, proved in [4], involving sequences of degenerate U-statistics of order 2. We stress that the results contained in [4] have later been extended to degenerate U-statistics of a general order; see [1, 5]. Albeit our method clearly applies to these general objects, we prefer here to focus on U-statistics of order 2, in order to obtain neater statements and to emphasize the method over technical details. We start with some useful definitions. Definition 2.10 (U-statistics). (i) Letk 2, andleth :Rq Rbeasymmetrickernelsuchthat ≥ → h L1 (µk). The(symmetric)U-statistic of order k basedonhandonthesample Y ,...,Y ∈ sym { 1 m} (where m k is some integer) is the random variable ≥ m (2.9) U (h,Y) = 6= h(Y ,...,Y ), m i1 ik i1,.X..,ik=1 where the symbol 6= indicates that the sum is taken over all vectors (i ,...,i ) such that 1 k i = i for every j = ℓ. j ℓ 6 6 P (ii) Fix k 2 and let U (h,Y) be a symmetric U-statistic as in (2.9). The Hoeffding rank of m ≥ U (h,Y) is the smallest integer 1 q k such that E[h(Y ,...,Y )Y ,...,Y ] = 0 (a.s.-P) m 1 k 1 q−1 ≤ ≤ | and E[h(Y ,...,Y )Y ,...,Y ] = 0, where E[h(Y ,...,Y )Y ,...,Y ] := E[h(Y ,...,Y )]. A U- 1 k 1 q 1 k 1 0 1 k | 6 | statistic of order k with Hoeffding rank equal to k is said to be completely degenerate. In other words, a U-statistic such as (2.9) is completely degenerate if h is a non-zero kernel verifying h(x,y ,...,y )p(x)dx = 0 (µk−1 a.e.). 1 k−1 R − Z (iii) A collection of random variables F : n 1 is said to be a sequence of geometric U-statistics n { ≥ } of order k, if there exists a kernel h L1 (µk) such that ∈ sym F = U (h,Y), n 1, n N(n) ≥ where N(n) :n 1 is the independent Poisson sequence introduced above. { ≥ } 7 Before presenting the main result of this section, and in order to make the connection with our general framework more transparent, we shall recall an important finding from [30, Lemma 3.5 and Theorem 3.6], stating that Poissonized U-statistics of order k live inside the sum of the first k + 1 Wiener chaoses associated with the Poisson measure η . The proof heavily relies on results by Last n and Penrose [12]. Lemma 2.11 (Reitzner and Schulte). Consider a kernel h L1 (µk) such that the corresponding ∈ sym Poissonized U-statistic U (h,Y) is square-integrable. Then, h is necessarily in L2 (µk), and N(n) sym U (h,Y) admits a chaotic representation of the type N(n) k U (h,Y) = E[U (h,Y)]+ nk−iI (h ) N(n) N(n) i i i=1 X where I indicates a multiple Wiener-Itô integral of order i with respect to the compensated Poisson i measure ηˆ = η µ , defined according to (2.8), and n n n − k (2.10) h (z ,...,z ) = h(z ,...,z , )µk−i(d ), (z ,...,z ) i, i 1 i 1 i 1 i i • • ∈ Z (cid:18) (cid:19)ZZk−i where the bullet “ ” stands for a packet of k i variables that are integrated with respect to µk−i. In • − particular, h = h and the projection h is in L2 (µi) for each 1 i k. k i sym ≤ ≤ Thefollowing statement correspondstothemainresultprovedbydeJongin[4],inthespecialcase of symmetric U-statistics of order 2 (note that the assumption that the underlying kernels have finite moments of order four is only implicit in de Jong’s work). Given positive sequences a ,b , n 1, we n n ≥ write a b whenever lim a /b = 1. n n n n ≈ n→∞ Theorem 2.12 (de Jong). Let h : n 1 be a sequence of non-zero elements of L4 (µ2). Define { n ≥ } sym F = U (h ,Y) and assume that F is completely degenerate. Then, one has that σ2(n):= Var(F ) n n n n n ≈ 2n2E[h (Y ,Y )2], and the fourth moment condition n 1 2 E[F4] lim n = 0, n→∞ σ(n)4 implies that, as n , the sequence F := F /σ(n) converges in distribution to a standard Gaussian n n → ∞ random variable. e The following statement consists of two parts. Part (A) is a quantitative extension of Theorem 2.12basedonadirectstudyofthefourthmomentsofthePoissonized U-statistic, whereaspart(B)isa Gamma-type extension of deJong’stheorem whichis directly basedontheresultsdiscussedinSection 2.1. Apart from [4], our findings should be compared with the seminal work by Jammalamadaka and Janson[8],aboutthenormalandPoissonapproximationofU-statisticsofordertwo. Toourknowledge, the forthcoming Theorem 2.13 is the first quantitative extensions of the de Jong theorem, also dealing with the non-normal approximation of general degenerate U-statistics. Moreover, we would like to emphasize that our proof of Part (A) is shorter and more transparent than the one presented in the original work [4](one should notethat, however, ourmethods only allow us to dealwith symmetric U- statistics). Recall that the Wasserstein distance between the laws of two integrable random variables X,Y is given by d (X,Y):= sup E[h(X)] E[h(Y)] , W | − | h∈Lip(1) where Lip(1) is the set of Lipschitz functions h : R R with a Lipschitz constant 1. Recall that, → ≤ in the framework of this section, = Rd. Z 8 Theorem 2.13 (Extended de Jong theorem). Let h : n 1 be a sequence of non-zero elements n { ≥ } of L4 (µ2) such that sym h4 dµ2 sup Z n n < . n RZh2ndµ2n 2 ∞ Put F = U (h ,Y) and F′ = U (h ,Y), and assume that these U-statistics are completely n n n n N(n) n(cid:0)R (cid:1) degenerate. Then, σ(n)2 := Var(F ) Var(F′) = 2n2E[h (Y ,Y )2], and the following two points (A) n ≈ n n 1 2 and (B) hold. (A) If E[(F′)4] (2.11) n 0 as n , σ(n)4 → → ∞ then both F := F /σ(n) and F′ := F′/σ(n) converge in distribution to a standard Gaussian n n n n random variable N. Moreover, there exists a universal finite constant K, independent of n, such that, as n e , e → ∞ (2.12) d (F′,N) K B 0, W n ≤ × n −→ (2.13) d (F ,N) K B +n−1/4 0, W n n ≤ e× −→ with Bn := σ(n)−2max Z h4nedµ2n 1/2;khn ⋆(cid:0)11hnk;khn ⋆(cid:1)12hnk . n o (B) If h4 dµ2 0 and ther(cid:0)eRexists ν >(cid:1) 0 such that σ(n)2 2ν, and Z n n → → (2.R14) E[(F′)4] 12E[(F′)3] 12ν2 48ν as n , n − n −→ − → ∞ then both F and F′ converge in distribution to a random variable G(ν), which has distribution n n Γ . Moreover, there exists a universal constant K > 0 such that, as n , ν → ∞ (2.15) d (F′,Γ ) c A (F′)+c A (F′) A (F′) K C 0, 3 n ν ≤ 1 1 n 2 4 n × 5 n ≤ × n −→ (2.16) d (F ,Γ ) K C +n−1/4 0, 3 n ν n ≤ × −→ with C := max 2 h 2 2ν ; h4 dµ2 1/4; h ⋆1h(cid:0) 1/2; h ⋆(cid:1)1h h , and we have n | k nk − | Z n n k n 2 nk k n 1 n − nk used the notationnintroduced in (2.1)–(2.3). o (cid:0)R (cid:1) e Remark 2.14. Our proof of Theorem 2.13 shows indeed that the quantity B (reps. C ) in the n n statement converges to zero if and only if the asymptotic condition (2.11) (resp. (2.14)) is verified. 2.5 Gamma convergence of geometric U-statistics: characterization and bounds As anticipated, the aim of this section is to apply the main estimates of the present paper in order to characterize the class of geometric U-statistics based on Y converging in distribution towards a Gammarandomvariable. Sinceouranalysisis basedonTheorem2.13, ourresultswillprovideexplicit estimates on the speed of convergence. We refer the reader to [6, 32] for some classic references on the subject and to [11, 30] for a discussion of several recent developments. We let the notation and assumptions of the previous section prevail and recall that a Gaussian measure G on Rd,B(R)⊗d , with control µ(dx) =p(x)dx, is a centred Gaussian family of the type (cid:0) (cid:1) G= G(B) :B B(R)⊗d, µ(B) < { ∈ ∞} such that, for every m 1 and every B ,...,B B(R)⊗d with µ(B ) < (i = 1,...,m), the 1 m i ≥ ∈ ∞ vector G(B ),...,G(B ) has an m-dimensional joint Gaussian distribution with covariance matrix 1 m E[G(B )G(B )] = µ(B B ). i j i j (cid:0) ∩ (cid:1) Thenextstatementcombinesfindingsfrom[11,Section7](point(i))withaclassiccharacterization of elements in the second Wiener chaos of a Gaussian measure (point (ii); see [18, Section 2.7.4] for more details. 9 Proposition 2.15. Let k 2 and let h L1 (µk) be a non-zero kernel such that the U-statistic ≥ ∈ sym F′ := U (h,Y) is square-integrable for every n, and has Hoeffding rank equal to 2. For n 1, n N(n) ≥ define also the standardized U-statistic F′ = n1−kF′. n n (i) For every n, there exists a sequence of double integrals I (f ) (each realized with respect to the 2 n compensated Poisson measure η eµ ) such that, as n , E[(F′ I (f ))2] 0. Moreover, n− n → ∞ n− 2 n → F′ converge in distribution to IG(h ), where IG indicates a double Wiener-Itoˆ integral with n 2 2 2 respect to the Gaussian measure G, and h is defined according to e(2.10). The same convergence 2 teakes place for the de-Poissonized U-statistics F = n1−kF , where F := U (h,Y). n n n n (ii) The random variable IG(h ) cannot be Gaussian. Moreover, assume that IG(h ) follows a 2 2 e 2 2 Γ -distribution. Then, necessarily, ν 1,2,... and there exists an orthonormal system ν ν ∈ { } e ,...,e L2(µ) such that h = e e and E[e (Y )] = 0 for i= 1,...,ν. 1 ν 2 i i i 1 { } ⊂ ⊗ i=1 X ν Remark 2.16. Let k = 2, and consider h = e e , as in the statement of Proposition 2.15-(ii). 2 i i ⊗ i=1 Then, it is easily seen (by a direct computationP) that ν n 2 n U (h ,Y) = e (Y ) e (Y )2 . n 2 i k i k  −  ! i=1 k=1 k=1 X X X   The fact that the distributions of F′ and F converge to Γ is therefore a direct consequence of the n n ν usual multidimensional central limit theorem and of the law of large numbers. e e The next statement is a quantitative counterpart to Proposition 2.15-(ii), containing in particular estimates involving Malliavin operators. Such estimates will be put into use in the Examples 2.22– 2.26 below, where the asymptotic behavior of a U-statistic such as U (h ,Y) is studied within the N(n) 2 framework of hybrid convergence in random graphs, random flat and random simplex models. Theorem 2.17 (Bounds on Gamma convergence). Let the assumptions and notation of Pro- position 2.15 prevail. Assume moreover that IG(h ) has distribution Γ for some ν = 1,2,..., and 2 2 ν also that e ,...,e L4(µ), where the orthonormal system e ,...,e is defined in Proposition 1 ν 1 ν { } ⊂ { } 2.15-(ii). Then, there exists a finite constant K, independent of n, such that d (F′,Γ ) c A (F˜′)+c A (F˜′) A (F˜′) K n−1/4, 3 n ν ≤ 1 1 n 2 4 n × 5 n ≤ × d (F ,Γ ) K n−1/4, 3 n ν e ≤ × where in the first inequality we used the notation defined in (2.1)–(2.3) e Example 2.18. (i) Let g ,g be two orthonormal elements of L2(µ) such that g ,g L4(µ) and 1 2 1 2 ∈ g (z)µ(dz) = g (z)µ(dz). We stress that we do not require that g ,g have disjoint Z 1 Z 2 1 2 supports. Then, the kernel R R 1 g (z ) g (z ) g (z ) g (z ) 1 1 2 1 1 2 2 2 h (z ,z ) = (g g ) (g g )(z ,z ) = − − 2 1 2 1 2 1 2 1 2 2 − ⊗ − √2 × √2 is such thatthe correspondingU-statistics of order two F := U (h ,Y)and F′ := U (h ,Y) n n 2 n N(n) 2 arecompletelydegenerate, andbothconvergeindistributiontoΓ ,withanupperboundoforder 1 n−1/4 on the rate of convergence. (ii) As an example of a pair (g ,g ) verifying the requirements at Point (i), one can take g = √21 1 2 1 A and g = √21 , where A,B is a measurable partition of such that µ(A) = µ(B) = 1/2. 2 B { } Z Considering the case d = 1, p( ) = 1 1 ( ), g (z) = √21 (z) and g (z) = √21 (z), · 2 (−1,1) · 1 (0,1) 2 (−1,0) oneobtains a kernel h with supportin( 1,1)2 (0,0) andsuch that h (z ,z )= 1 if z z >0, 2 2 1 2 1 2 − \{ } and h (z ,z ) = 1 if z z < 0. In this way, one recovers the non-central result discussed by 2 1 2 1 2 − Reitzner and Schulte in [30, end of Section 5.1]. 10