Central limit theorems for U -statistics of Poisson point processes
aa r X i v : . [ m a t h . P R ] D ec The Annals of Probability (cid:13)
Institute of Mathematical Statistics, 2013
CENTRAL LIMIT THEOREMS FOR U -STATISTICS OF POISSONPOINT PROCESSES By Matthias Reitzner and Matthias Schulte
University of Osnabrueck and Karlsruher Institut f¨ur Technologie A U -statistic of a Poisson point process is defined as the sum P f ( x , . . . , x k ) over all (possibly infinitely many) k -tuples of dis-tinct points of the point process. Using the Malliavin calculus, theWiener–Itˆo chaos expansion of such a functional is computed andused to derive a formula for the variance. Central limit theoremsfor U -statistics of Poisson point processes are shown, with explicitbounds for the Wasserstein distance to a Gaussian random variable.As applications, the intersection process of Poisson hyperplanes andthe length of a random geometric graph are investigated.
1. Introduction.
In recent years, Malliavin calculus, Wiener–Itˆo chaosexpansions and Fock space representations of functionals of Poisson pointprocesses have been a rapidly developing topic. First results already ap-peared in the classical works of Itˆo [13, 14] and Wiener [37]. Yet only inthe last years prominent contributions produced a deep theory which mostprobably will have a strong impact on modern theory and applications ofPoisson point processes; see, for example, Houdre and Perez-Abreu [12], Lastand Penrose [19], Nualart and Vives [25] and Wu [38]. Here in particular wewant to point out the groundbreaking paper by Peccati et al. [27] on centrallimit theorems using Stein’s method and Malliavin calculus. These methodswere combined the first time by Nourdin and Peccati [24] for functionals de-pending on Gaussian processes instead of Poisson point processes. Furtherdevelopments include the book of Peccati and Taqqu [28] about productformulas for multiple Wiener–Itˆo integrals in the Gaussian and Poisson caseand a central limit theorem due to Peccati and Zheng [29] generalizing themain result of [27] to random vectors.Poisson point processes occur in many branches of probability theory,for example, in the theory of Levy processes, and in the theory of random
Received May 2011; revised July 2012.
AMS 2000 subject classifications.
Primary 60H07, 60F05; secondary 60G55, 60D05.
Key words and phrases.
Central limit theorem, Malliavin calculus, Poisson point pro-cess, Stein’s method, U -statistic, Wiener–Itˆo chaos expansion. This is an electronic reprint of the original article published by theInstitute of Mathematical Statistics in
The Annals of Probability ,2013, Vol. 41, No. 6, 3879–3909. This reprint differs from the original inpagination and typographic detail. 1
M. REITZNER AND M. SCHULTE graphs, in spatial statistics, in communication theory and in stochastic ge-ometry. Hence there is a wide range of potential applications of these newresults. In this work, we use the Wiener–Itˆo chaos expansion and a relatedresult from [27] to prove central limit theorems for a broad class of function-als, namely for U -statistics of Poisson point processes.Let η be a Poisson point process over a state space X . We call a randomvariable F a U -statistic of η if F ( η ) = X ( x ,...,x k ) ∈ η k = f ( x , . . . , x k ) . By η k = we denote the set of all k -tuples of distinct points of the process.One should compare this definition to classical U -statistics defined on a setof n random variables { Z , . . . , Z n } = ζ where U ( ζ ) = P ζ k = f ( x , . . . , x k ). Fordetails on classical U -statistics we refer to [11, 16, 20]. From now on, wemean by U -statistic a U -statistic of a Poisson point process.The first step in this paper is the explicit evaluation of expressions involv-ing Malliavin operators acting on U -statistics of Poisson point processes. Themain result of this paper is Theorem 4.7 which gives an explicit bound onthe Wasserstein distance between a normalized U -statistic and a standardGaussian random variable N , d W (cid:18) F − E F √ Var
F , N (cid:19) ≤ k / X ≤ i ≤ j ≤ k p M ij ( f )Var F , where M ij ( f ) are sums of certain fourth moment integrals. If the intensitymeasure of η is of the form µ = λθ with an intensity parameter λ ≥ θ , one is interested in the behavior of F for increasing λ . In theparticular situation that f : X k → R is independent of λ , we conclude inTheorem 5.2 that d W (cid:18) F − E F √ Var
F , N (cid:19) ≤ C f λ − / . In general this is the optimal rate in λ because for a set A ⊂ X with θ ( A ) = 1the U -statistic F = P x ∈ η ( x ∈ A ) is Poisson distributed with parameter λ ,and it is widely known that a Poisson distributed random variable has thisrate of convergence.As an application of our result we investigate the intrinsic volumes of theintersection process of Poisson hyperplanes in a compact convex window.A central limit theorem for some of these functionals was proved in two longand intricate papers by Heinrich [9] and Heinrich, Schmidt and Schmidt [10].Here we obtain a general result which in addition gives rates of convergenceto Gaussian variables. A second example concerns functionals of Sylvester LT’S FOR POISSON U -STATISTICS type by which we mean the question about the probability that k points ina convex set are in convex position. Our last example is about the number ofedges of a random geometric graph in a bounded window. Again we obtaina central limit theorem with a rate of convergence. As general references tostochastic geometry and random graphs we refer to [30, 32] and [35].To prove our central limit theorems, we first use a result of Last andPenrose [19], to expand a U -statistic in a Wiener–Itˆo chaos expansion as afinite sum of multiple Wiener–Itˆo integrals. This enables us to give a formulafor the variance of a U -statistic and to compute two operators from Malliavincalculus that are defined by their chaos expansions. Using a theorem forthe normal approximation of Poisson functionals due to Peccati et al. [27],we show convergence in the Wasserstein distance. In order to apply theirresult, we need to compute expected values of products of multiple Wiener–Itˆo integrals which is well known to be a notorious difficult task. We expectthat the same techniques can be used to show central limit theorems formore general functionals of Poisson point processes.This paper is organized in the following way. In Section 2, we introduceWiener–Itˆo chaos expansions for functionals of a Poisson point process andsome operators from Malliavin calculus. Then we compute the Wiener–Itˆochaos expansion of a U -statistic and its variance in Section 3. Using Mallia-vin calculus we prove the general version of our central limit theorem for U -statistics in Section 4. Finally, we investigate two special classes of U -statistics and present examples in the Sections 5 and 6.
2. Wiener–Itˆo chaos expansions for Poisson point processes.
Poisson point process.
In this paper, we let η be a Poisson pointprocess on the measure space ( X, B ( X ) , µ ) where X is a Borel space and µ is a σ -finite nonatomic Borel measure. A Borel space is a measurable spacewhich is isomorphic to a Borel subset of [0 , , F , P ) be a probability space. Denote by N ( X )the set of all integer-valued σ -finite measures ν on X , equipped with thesmallest σ -algebra N ( X ) such that the mappings ν → ν ( A ) are measurablefor all sets A ∈ B ( X ). A random measure η : Ω → N ( X ) is called a Poissonpoint process with intensity measure µ if for A ∈ B ( X ) the random variable η ( A ) is Poisson distributed with parameter µ ( A ), and the random variables η ( A ) , . . . , η ( A m ) are independent for pairwise disjoint sets A , . . . , A m ∈B ( X ). Since the intensity measure µ is nonatomic, the Poisson point processis simple, that is, η ( { x } ) ≤ x ∈ X almost surely. Thus, we can view η as a random set of points in X .As usual, L p ( X k ) denotes the space of all measurable functions f : X k → R := R ∪ {±∞} with Z X k | f ( x , . . . , x k ) | p dµ ( x , . . . , x k ) < ∞ , M. REITZNER AND M. SCHULTE where dµ ( x , . . . , x k ) stands for dµ ( x ) · · · dµ ( x k ). Let L ps ( X k ) be the subsetof µ k -almost everywhere symmetric functions in L p ( X k ). We call a functionsymmetric if it is invariant under all permutations of its arguments. Wedenote by k · k the norm in L ( X k ), and by h· , ·i the inner product in L ( X k ).Equipped with this inner product, L ( X k ) and L s ( X k ) form Hilbert spaces.Instead of the original probability measure P , we always use the imagemeasure P = P ◦ η . In the following, L p ( P ) stands for the set of all measurablefunctions F : N ( X ) → R with E | F | p < ∞ .An important property of Poisson point processes is the Slivnyak–Meckeformula (see Corollary 3.2.3 in [32]) which says that E X ( x ,...,x k ) ∈ η k = f ( x , . . . , x k ) = Z X k f ( x , . . . , x k ) dµ ( x , . . . , x k )(1)for f ∈ L ( X k ). (Recall the definition of η k = in the Introduction.) The sumon the left-hand side is a priori defined as an L ( P ) limit summing onlyover points in an increasing window. Yet it follows from the Slivnyak–Meckeformula that f ∈ L ( X k ) implies that the sum on the left-hand side is abso-lutely convergent almost surely.2.2. Multiple Wiener–Itˆo integrals.
Now we present the definition of mul-tiple Wiener–Itˆo integrals of order k ∈ N following [36]. One starts with sim-ple functions and extends the definition to arbitrary functions in L s ( X k ).A function f ∈ L ( X k ) is called simple if:(1) f is symmetric;(2) f is constant on a finite number of Cartesian products B × · · · × B k ∈B ( X ) k and vanishes elsewhere;(3) f vanishes on diagonals, which means f ( x , . . . , x k ) = 0 if x i = x j forsome i = j .Let S ( X k ) be the space of all simple functions. For f ∈ S ( X k ) and k ∈ N , themultiple Wiener–Itˆo integral I k ( f ) of f with respect to the compensatedPoisson point process η − µ is defined by I k ( f ) = Z X k f d ( η − µ ) k = X f B ×···× B k ( η − µ )( B ) · · · ( η − µ )( B k ) , where we sum over all Cartesian products and f B ×···× B k is the value of f on such a set. For k = 0 we put I ( f ) = f . By a straightforward computation,one shows E I k ( f ) = k ! k f k . (2)Thus there is an isometry between S ( X k ) and a subset of L ( P ). Further-more, S ( X k ) is dense in L s ( X k ), whence for every f ∈ L s ( X k ) there is asequence ( f n ) n ∈ N of simple functions with f n → f in L s ( X k ). Because of LT’S FOR POISSON U -STATISTICS the isometry (2), it is possible to define I k ( f ) as the limit of ( I k ( f n )) n ∈ N in L ( P ). Hence for an arbitrary symmetric function f ∈ L s ( X k ) we put f ( x , . . . , x k ) = f ( x , . . . , x k ) if x i = x j for all i = j and f ( x , . . . , x k ) = 0otherwise and obtain I k ( f ) = Z X k f d ( η − µ ) k . We remark that the denseness of S ( X k ) in L s ( X k ) depends on the topo-logical structure of X and the fact that µ is nonatomic. For a definitionwithout these requirements we refer to [19].It follows directly from the definition that multiple Wiener–Itˆo integralshave the properties summarized in the following: Lemma 2.1.
Let f ∈ L s ( X n ) and g ∈ L s ( X m ) with n, m ≥ . Then: (a) E I n ( f ) = 0 ; (b) E I n ( f ) I m ( g ) = ( n = m ) n ! h f, g i . Wiener–Itˆo chaos expansions.
For a measurable function F : N ( X ) → R and y ∈ X we define the difference operator as D y F ( η ) = F ( η + δ y ) − F ( η ) , where δ y is the Dirac measure at the point y . The difference operator D y F measures the effect of adding the point y ∈ X to the Poisson point process,whence it is also denoted as add one cost operator in [19]. The iterateddifference operator is defined by D y ,...,y i F = D y D y ,...,y i F. Let the functions f i : X i → R be given by f = E F and f i ( y , . . . , y i ) = 1 i ! E D y ,...,y i F, i ≥ , if these expectations exist. Because of the symmetry of the iterated differenceoperator, f i is symmetric if defined. The following relationships between F ,the functions f i , i ∈ N , and the variance of F have been shown by Last andPenrose [19]. Theorem 2.2 (Last and Penrose [19]).
Let F ∈ L ( P ) . Then f i ∈ L s ( X i ) , i ∈ N and F = ∞ X i =0 I i ( f i ) , where the sum converges in L ( P ) . The f i ∈ L s ( X i ) , i ∈ N are the µ i -almosteverywhere unique g i ∈ L s ( X i ) , i ∈ N , satisfying F = P ∞ i =0 I i ( g i ) in L ( P ) . M. REITZNER AND M. SCHULTE
Furthermore,
Var F = ∞ X i =1 i ! k f i k . In the following, we call the functions f i , i ∈ N , kernels of the Wiener–Itˆochaos expansion of F . The class of sequences ( g i ) i ∈ N with g i ∈ L s ( X i ) and ∞ X i =0 i ! k g i k < ∞ composes a Hilbert space isomorphic to the symmetric Fock space associatedwith L ( X ). In this context, Theorem 2.2 states that there exists an isometrybetween L ( P ) and a symmetric Fock space.2.4. Malliavin calculus.
Our proofs for central limit theorems are basedon a result for the normal approximation of Poisson functionals from [27],which uses operators from Malliavin calculus. In the following, we give ashort introduction to these operators. For more details we refer to [19, 25, 27].Let F ∈ L ( P ) and f i , i ∈ N , be the kernels of the Wiener–Itˆo chaos ex-pansion of F . First of all, we give an alternative definition of the differenceoperator D y using the Wiener–Itˆo chaos expansion of F . Definition 2.3.
Let ∞ X i =1 ii ! k f i k < ∞ . (3)Then the random function y D y F, y ∈ X , is given by D y F = ∞ X i =1 iI i − ( f i ( y, · )) . It can be proved (see [25], Theorem 6.2 or [19], Theorem 3.3) that for F ∈ L ( P ) satisfying (3) this definition coincides with the one introduced inSection 2.3. Definition 2.4. If ∞ X i =1 i i ! k f i k < ∞ , then the Ornstein–Uhlenbeck generator LF is the random variable given by LF = − ∞ X i =1 iI i ( f i ) . LT’S FOR POISSON U -STATISTICS The Ornstein–Uhlenbeck generator has an inverse operator. Its domain isthe space of all centred F ∈ L ( P ), that is, F ∈ L ( P ) with E F = 0, and L − F = − ∞ X i =1 i I i ( f i ) . If F is in the domain of L , then the Ornstein–Uhlenbeck generator canbe written as LF = Z X F ( η − δ x ) − F ( η ) dη ( x ) − Z X ( F ( η ) − F ( η + δ z )) dµ ( z ) . (4)This follows from the representation of the difference operator and theSkorohod-integral (see [19], formula (3.19)), which is not used in this work.
3. Malliavin calculus and Wiener–Itˆo chaos expansions for U -statistics. In this section, we define U -statistics of Poisson point processes and investi-gate their Wiener–Itˆo chaos expansions. In particular, we apply the Malliavinoperators to U -statistics and present explicit formulae for the kernels of theWiener–Itˆo chaos expansion and the variance.3.1. U -statistics of Poisson point processes. Recall the definition η k = = { ( x , . . . , x k ) ∈ η k , x i = x j for i = j } from the Introduction. Definition 3.1.
A random variable F = X ( x ,...,x k ) ∈ η k = f ( x , . . . , x k )(5)with f ∈ L s ( X k ) is called U -statistic of order k .By the Slivnyak–Mecke formula (1), it holds that E X ( x ,...,x k ) ∈ η k = f ( x , . . . , x k ) = Z X · · · Z X f ( x , . . . , x k ) dµ ( x , . . . , x k )so that f ∈ L s ( X k ) guarantees F ∈ L ( P ). Due to the fact that we sum overall permutations of k points in (5), we can assume without loss of generalityin Definition 3.1 that f is symmetric.Since we want to use Wiener–Itˆo chaos expansions, we always require that F is in L ( P ). For the central limit theorems we additionally assume that F is absolutely convergent. Definition 3.2. A U -statistic F is absolutely convergent if F = X ( x ,...,x k ) ∈ η k = | f ( x , . . . , x k ) | is in L ( P ). M. REITZNER AND M. SCHULTE
Note that F absolutely convergent implies that F ∈ L ( P ). Obviouslyevery F ∈ L ( P ) with f ≥ Malliavin calculus.
We start by calculating the difference operatorof a U -statistic F . Lemma 3.3.
Let F ∈ L ( P ) be a U -statistic of order k . Then the differ-ence operator applied to F gives D y F = k X ( x ,...,x k − ) ∈ η k − = f ( y , x , . . . , x k − ) . Proof.
By the definition of the difference operator D y and the symme-try of f , we obtain for a U -statistic D y F = X ( x ,...,x k ) ∈ ( η ∪{ y } ) k = f ( x , . . . , x k ) − X ( x ,...,x k ) ∈ η k = f ( x , . . . , x k )= X ( x ,...,x k − ) ∈ η k − = ( f ( y , x , . . . , x k − ) + · · · + f ( x , . . . , x k − , y ))= k X ( x ,...,x k − ) ∈ η k − = f ( y , x , . . . , x k − ) . (cid:3) An analogous straightforward computation using (4) verifies the followinglemma.
Lemma 3.4.
Let F ∈ L ( P ) be a U -statistic of order k . Then the Ornstein–Uhlenbeck operator applied to F gives LF = − kF + k Z X X ( x ,...,x k − ) ∈ η k − = f ( x , . . . , x k − , z ) dµ ( z ) . Without proof we also state the inverse Ornstein–Uhlenbeck operator ofa U -statistic. L − ( F − E F )= k X m =1 m ! Z X k f ( y , . . . , y k ) dµ ( y , . . . , y k ) − k X m =1 m X ( x ,...,x m ) ∈ η m = Z X k − m f ( x , . . . , x m , y , . . . , y k − m ) dµ ( y , . . . , y k − m ) . LT’S FOR POISSON U -STATISTICS Wiener–Itˆo chaos expansions.
Let us now compute the kernels andthe Wiener–Itˆo chaos expansion of a U -statistic F = P η k = f with F ∈ L ( P ). Lemma 3.5.
Let F ∈ L ( P ) be a U -statistic of order k . Then the kernelsof the Wiener–Itˆo chaos expansion of F have the form f i ( y , . . . , y i ) = (cid:18) ki (cid:19) Z X k − i f ( y , . . . , y i , x , . . . , x k − i ) dµ ( x , . . . , x k − i ) ,i ≤ k, , i > k, and F has the variance Var F = k X i =1 i ! (cid:18) ki (cid:19) × Z X i (cid:18)Z X k − i f ( y , . . . , y i , x , . . . , x k − i ) dµ ( x , . . . , x k − i ) (cid:19) (6) dµ ( y , . . . , y i ) . For the special case k = 2 the formulas for the kernels are already implicitin the paper by Molchanov and Zuyev [22] where ideas closely related toMalliavin calculus have been used. Proof of Lemma 3.5.
In Lemma 3.3, the difference operator of a U -statistic was computed. Proceeding by induction, we get D y ,...,y i F = k !( k − i )! X ( x ,...,x k − i ) ∈ η k − i = f ( y , . . . , y i , x , . . . , x k − i )for i ≤ k . Hence D y ,...,y k F only depends on y , . . . , y k and is independent ofthe Poisson point process. This yields D y ,...,y k +1 F = 0 and D y ,...,y i F = 0for all i > k . We just proved D y ,...,y i F = k !( k − i )! X ( x ,...,x k − i ) ∈ η k − i = f ( y , . . . , y i , x , . . . , x k − i ) , i ≤ k, , otherwise. M. REITZNER AND M. SCHULTE
By the Slivnyak–Mecke formula (1), we obtain f i ( y , . . . , y i ) = 1 i ! E D y ,...,y i F = 1 i ! E k !( k − i )! X ( x ,...,x k − i ) ∈ η k − i = f ( y , . . . , y i , x , . . . , x k − i )= k ! i !( k − i )! Z X k − i f ( y , . . . , y i , x , . . . , x k − i ) dµ ( x , . . . , x k − i )for i ≤ k . The formula for the variance follows from Proposition 2.2. (cid:3) Note that F ∈ L ( P ) implies f i ∈ L s ( X i ), and thus that for all 1 ≤ i ≤ k Z X i (cid:18)Z X k − i f ( y , . . . , y i , x , . . . , x k − i ) dµ ( x , . . . , x k − i ) (cid:19) dµ ( y , . . . , y i ) < ∞ . In particular, it holds f ∈ L s ( X k ).By Lemma 3.5, U -statistics only have a finite number of nonvanishingkernels. The following theorem characterizes a U -statistic by this property.We call a Wiener–Itˆo chaos expansion finite if only a finite number of kernelsdo not vanish. Theorem 3.6.
Assume F ∈ L ( P ) . (1) If F is a U -statistic, then F has a finite Wiener–Itˆo chaos expansionwith kernels f i ∈ L s ( X i ) ∩ L s ( X i ) , i = 1 , . . . , k . (2) If F has a finite Wiener–Itˆo chaos expansion with kernels f i ∈ L s ( X i ) ∩ L s ( X i ) , i = 1 , . . . , k , then F is a (finite) sum of U -statistics and a constant. Proof.
The fact that a U -statistic F ∈ L ( P ) has a finite Wiener–Itˆochaos expansion with f i ∈ L s ( X i ) follows from Lemma 3.5 and from f ∈ L s ( X k ).For the second part of the proof, let F ∈ L ( P ) have a finite Wiener–Itˆochaos expansion, that is, F = m X i =0 I i ( f i )with kernels f i ∈ L s ( X i ) ∩ L s ( X i ) and m ∈ N . Now Proposition 4.1 in [36]implies that I i ( f i ) = i X j =0 ( − i − j (cid:18) ij (cid:19) X ( x ,...,x j ) ∈ η j = f ( j ) i ( x , . . . , x j ) , LT’S FOR POISSON U -STATISTICS where the inner sum is a constant for j = 0 and f ( j ) i is given by f ( j ) i ( x , . . . , x j ) = Z X i − j f i ( x , . . . , x j , y , . . . , y i − j ) dµ ( y , . . . , y i − j ) . The assumption f i ∈ L s ( X i ) guarantees f ( j ) i ∈ L s ( X j ) for j = 1 , . . . , i and f (0) i ∈ R . Hence, every Wiener–Itˆo integral is a (finite) sum of U -statisticsand a constant, and the same holds for F . (cid:3) Examples.
The following examples show that the assumptions on F and f i in Theorem 3.6 are necessary. In all examples, we consider a Poissonpoint process in R with the Lebesgue measure as intensity measure. Example.
There exist random variables in L ( P ) with finite Wiener–Itˆo chaos expansions which are not sums of U -statistics. This is possible ifthe kernels f i are in L s ( X i ) \ L s ( X i ). Define g : R → R as g ( x ) = 1 x ( | x | > , which is in L ( R ) \ L ( R ). Now we define the random variable G = I ( g ). G is in L ( P ) and has a finite Wiener–Itˆo chaos expansion. But the formalrepresentation I ( g ) = X x ∈ η g ( x ) − Z R g ( x ) dx we used in the proof of Theorem 3.6 fails because the integral does not exist. Example.
There also exist U -statistics F ∈ L ( P ) with f ∈ L s ( X k ) ∩ L s ( X k ) which are not in L ( P ). We construct f ∈ L s ( R ) ∩ L s ( R ) with k f k = ∞ by putting f ( x , x ) = (0 ≤ x √ x ≤ (0 ≤ x √ x ≤ F = X ( x ,x ) ∈ η = f ( x , x ) . In this case the first kernel, f ( y ) = E (cid:20) X x ∈ η f ( y, x ) (cid:21) = 2 Z R f ( y, x ) dx = 2 ( y ≥
0) min (cid:26) y , √ y (cid:27) is not in L s ( R ) so that F has no Wiener–Itˆo chaos expansion and cannot bein L ( P ). M. REITZNER AND M. SCHULTE
Example.
By Theorem 3.6(2), a functional F ∈ L ( P ) with a finiteWiener–Itˆo chaos expansion and kernels f i ∈ L s ( X i ) ∩ L s ( X i ), i = 1 , . . . , k ,is a (finite) sum of U -statistics. Our next example shows that neither thesingle U -statistics are in L ( P ) nor are the summands necessarily in L s ( X i ).Set F = I ( f ) with f as above. Then I ( f ) = Z R f ( x, y ) dx dy − X x ∈ η Z R f ( x, y ) dy + X ( x ,x ) ∈ η = f ( x , x ) , and F is a sum of U -statistics. Since E [( P x ∈ η R R f ( x, y ) dy ) ] = ∞ , we knowthat the U -statistic X x ∈ η Z R f ( x, y ) dy is not in L ( P ), nor are the summands R R f ( x, y ) dy in L ( R ). This is incontrast to the remark after the proof of Lemma 3.5 that for a U -statistic F ∈ L ( P ), we always have f ∈ L ( X k ). Example.
To motivate the definition of an absolutely convergent U -statistic, we give an example of a U -statistic that is in L ( P ) but not abso-lutely convergent. Similarly to the previous examples, we set f ( x , x ) = (0 ≤ | x | p | x | ≤ (0 ≤ | x | p | x | ≤ ( x x ≥ − F = X ( x ,x ) ∈ η = f ( x , x ) and F = X ( x ,x ) ∈ η = | f ( x , x ) | . Now it is easy to verify that f ( x ) = 0 and f ( x , x ) = f ( x , x ) so that F ∈ L ( P ). But the first kernel of F is not in L ( R ) so that F / ∈ L ( P ).
4. Central limit theorems for U -statistics. In this section, we derive acentral limit theorem for U -statistics of Poisson point processes. In particu-lar, we are interested in the Wasserstein distance of a normalized U -statisticand a standard Gaussian random variable. Recall that the Wasserstein dis-tance d W ( Y, Z ) of two random variables Y and Z is given by d W ( Y, Z ) = sup h ∈ Lip(1) | E h ( Y ) − E h ( Z ) | , where Lip(1) is the set of all functions h : R → R with a Lipschitz-constantless than or equal to one. It is important to note that convergence in theWasserstein distance implies convergence in distribution. In particular, it isknown (see [4], e.g.) that for a Gaussian random variable N we have | P ( Y ≤ t ) − P ( N ≤ t ) | ≤ p d W ( Y, N )for all t ∈ R . Hence, we can prove central limit theorems by showing conver-gence to a Gaussian random variable in the Wasserstein distance. LT’S FOR POISSON U -STATISTICS Our main estimate for the distance between F = P η k = f and a standardGaussian random variable N is Theorem 4.7 which states that d W (cid:18) F − E F √ Var
F , N (cid:19) ≤ k / X ≤ i ≤ j ≤ k p M ij ( f )Var F , where the M ij ( f ) are sums of certain fourth moment integrals. The precisedefinition is given in formula (14). In most applications, it is elementary tobound these fourth moments of f . This is carried out in Sections 5 and 6.4.1. An abstract CLT.
Our most general result is the following upperbound for the Wasserstein distance of a Poisson functional with a finiteWiener–Itˆo chaos expansion and a standard Gaussian random variable. Toneatly formulate our results and proofs, we use the abbreviations R ij = E (cid:18)Z X I i − ( f i ( z, · )) I j − ( f j ( z, · )) dµ ( z ) (cid:19) (7) − (cid:20) E Z X I i − ( f i ( z, · )) I j − ( f j ( z, · )) dµ ( z ) (cid:21) , ˜ R i = E Z X I i − ( f i ( z, · )) dµ ( z )(8)for i, j = 1 , . . . , k . Note that R = 0 and that for i = j the second expectationin R ij vanishes. Theorem 4.1.
Suppose F ∈ L ( P ) has a finite Wiener–Itˆo chaos expan-sion of order k , and N is a standard Gaussian random variable. Then d W (cid:18) F − E F √ Var
F , N (cid:19) ≤ k X ≤ i,j ≤ k p R ij Var F + k / k X i =1 p ˜ R i Var F (9) with R ij and ˜ R i defined in (7) and (8). Proof.
Our proof is based on the following result of Peccati et al. (The-orem 3.1 in [27]), which is derived by a combination of Malliavin calculusand Stein’s method.
Theorem 4.2 (Peccati et al. [27]).
Let G ∈ L ( P ) with E G = 0 be in thedomain of D and let N be a standard Gaussian random variable. Then d W ( G, N ) ≤ E | − h DG, − DL − G i| + Z X E [ | D z G | | D z L − G | ] dµ ( z ) ≤ q E (1 − h DG, − DL − G i ) + Z X E [ | D z G | | D z L − G | ] dµ ( z ) . M. REITZNER AND M. SCHULTE
From now on, we denote by G = F − E F √ Var F the normalization of F and by g i ∈ L s ( X i ) , i = 1 , . . . , k , the kernels of G .Thus, it follows g i ( x , . . . , x i ) = 1 √ Var
F f i ( x , . . . , x i )for i = 1 , . . . , k and Var G = P ki =1 i ! k g i k = 1.Since F has a finite Wiener–Itˆo chaos expansion, F is in the domain of D , and we can apply the above theorem. By the definitions of the Malliavinoperators and the triangle inequality, we obtain E | − h DG, − DL − G i| = E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k X i =1 i ! k g i k − Z X k X i =1 iI i − ( g i ( z, · )) k X i =1 I i − ( g i ( z, · )) dµ ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k X i =2 E (cid:12)(cid:12)(cid:12)(cid:12) i ! k g i k − i Z X I i − ( g i ( z, · )) I i − ( g i ( z, · )) dµ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) + k X i,j =1 ,i = j i E (cid:12)(cid:12)(cid:12)(cid:12)Z X I i − ( g i ( z, · )) I j − ( g j ( z, · )) dµ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) . The first sum on the right-hand side of the inequality starts with i = 2 sincethe summand for i = 1 vanishes. As a consequence of Fubini’s theorem andLemma 2.1, it holds that E i Z X I i − ( g i ( z, · )) I i − ( g i ( z, · )) dµ ( z ) = i ! k g i k . Combining this with the Cauchy–Schwarz inequality leads to E (cid:12)(cid:12)(cid:12)(cid:12) i ! k g i k − i Z X I i − ( g i ( z, · )) I i − ( g i ( z, · )) dµ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ s E (cid:18) i ! k g i k − i Z X I i − ( g i ( z, · )) I i − ( g i ( z, · )) dµ ( z ) (cid:19) = s i E (cid:18)Z X I i − ( g i ( z, · )) I i − ( g i ( z, · )) dµ ( z ) (cid:19) − ( i !) k g i k = i √ R ii Var F LT’S FOR POISSON U -STATISTICS and E (cid:12)(cid:12)(cid:12)(cid:12)Z X I i − ( g i ( z, · )) I j − ( g j ( z, · )) dµ ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ s E (cid:18)Z X I i − ( g i ( z, · )) I j − ( g j ( z, · )) dµ ( z ) (cid:19) = p R ij Var F for i = j . Now it holds that E | − h DG, − DL − G i| ≤ k X i =2 i √ R ii Var F + k X i,j =1 ,i = j i p R ij Var F (10) ≤ k X ≤ i,j ≤ k p R ij Var
F .
Furthermore, again by the Cauchy–Schwarz inequality we have Z X E [( D z G ) | D z L − G | ] dµ ( z ) ≤ (cid:18)Z X E [( D z G ) ] dµ ( z ) (cid:19) / (cid:18)Z X E [( D z L − G ) ] dµ ( z ) (cid:19) / . By the definitions of the Malliavin operators and H¨older’s inequality, we canrewrite the expressions on the right-hand side as Z X E [( D z L − G ) ] dµ ( z ) = Z X k X i =1 E [ I i − ( g i ( z, · )) ] dµ ( z )= k X i =1 ( i − k g i k ≤ Z X E [( D z G ) ] dµ ( z ) ≤ Z X k k X i =1 i E [ I i − ( g i ( z, · )) ] dµ ( z ) = k k X i =1 i ˜ R i (Var F ) . Hence Z X E [( D z G ) | D z L − G | ] dµ ( z ) ≤ vuut k k X i =1 i ˜ R i (Var F ) (11) ≤ k √ k k X i =1 i p ˜ R i Var F ≤ k / k X i =1 p ˜ R i Var
F . M. REITZNER AND M. SCHULTE
Combining Theorem 4.2 with formulas (10) and (11) gives the right-handside of (9) in Theorem 4.1. (cid:3)
Estimates for the error terms.
To estimate the right-hand side of(9) for a U -statistic F = P η k = f in terms of the function f , we are interestedin the behavior of R ij and ˜ R i for i, j = 1 , . . . , k . Thus we need to computeexpected values of the type E m Y l =1 I n l ( f l ) , m ∈ N , n , . . . , n m ∈ N with f l ∈ L s ( X n l ) ∩ L s ( X n l ) for l = 1 , . . . , m . Such products of multipleWiener–Itˆo integrals are discussed in [28] and [36]. Before stating a resultfor the expected value of such a product, we introduce some notation. Thefunction N ml =1 f l : X P n l → R is given by m O l =1 f l ! ( z (1)1 , . . . , z (1) n , . . . , z ( m )1 , . . . , z ( m ) n m ) = m Y l =1 f l ( z ( l )1 , . . . , z ( l ) n l ) . Definition 4.3.
Let Π n ,...,n m be the set of all partitions of the set ofvariables z (1)1 , . . . , z ( m ) n m such that two variables z ( l ) i and z ( l ) j with i = j butthe same upper index ( l ) are always in different blocks, and such that everyblock includes at least two variables.In this definition, we think of variables as combinatorial objects and par-tition a set of them. This is slightly different from the approach in [28],where the variables are numbered, and the partitions are defined for a set ofnumbers. Observe that by definition each block of π ∈ Π n ,...,n m has at leasttwo and at most m variables each of them with different upper index ( l ).Subsequently also the following subset of Π n ,...,n m will play a central role. Definition 4.4.
Let Π n ,...,n m be the set of all partitions π ∈ Π n ,...,n m such that for any decomposition of { , . . . , m } into two disjoint nonemptysets M , M there are l ∈ M , l ∈ M and two variables z ( l ) i , z ( l ) j which arein the same block of π .By | π | we denote the number of blocks of the partition π . For every parti-tion π ∈ Π n ,...,n m we define the function ( N ml =1 f l ) π : X | π | → R by replacingall variables of N ml =1 f l that belong to the same block of π by a new com-mon variable. The order of the new variables does not matter since we alwaysintegrate over all variables.Let us recall that S ( X k ) stands for the set of simple functions. Theseare all f ∈ L s ( X k ) that are zero on all diagonals, are constant on a finite LT’S FOR POISSON U -STATISTICS number of Cartesian products, and vanish everywhere else. For the productof multiple Wiener–Itˆo integrals of such functions the following propositionholds; see Corollary 7.2 in [28]. Proposition 4.5.
Let f l ∈ S ( X n l ) for l = 1 , . . . , m . Then E m Y l =1 I n l ( f l ) = X π ∈ Π n ,...,nm Z X | π | m O l =1 f l ! π ( y , . . . , y | π | ) dµ ( y , . . . , y | π | ) . (12)As a consequence of Proposition 3.1 in [36], equation (12) is also true for f l ∈ L s ( X k ), l = 1 , . . . , m , satisfying m O l =1 f l ! π ∈ L ( X | π | )(13)for all partitions π of the set of variables such that all variables of a functionare in different blocks. For some classes of functions f l it is obvious that(13) holds, for example, if the f l are bounded and have a support of finitemeasure. But in general it is difficult to verify condition (13).In order to avoid this problem, we approximate a general U -statistic bya sequence of U -statistics, whose kernels are simple functions, and applyProposition 4.5. Afterward, we extend our results to the original U -statistic.From now on, we assume that F is an absolutely convergent U -statistic.Because of f ∈ L s ( X k ), there exists a sequence ( f ( n ) ) n ∈ N of functions in S ( X k ) such that | f ( n ) | ≤ | f | µ k -almost everywhere and ( f ( n ) ) n ∈ N convergesto f µ k -almost everywhere on X k . We define U -statistics F ( n ) , n ∈ N , by F ( n ) = X ( x ,...,x k ) ∈ η k = f ( n ) ( x , . . . , x k ) . Since ( f ( n ) ) n ∈ N converges µ k -almost everywhere on X k to f ,lim n →∞ f ( n ) ( x , . . . , x k ) = f ( x , . . . , x k ) for all ( x , . . . , x k ) ∈ η k = holds with probability 1. Furthermore, the absolute convergence of F implies | F ( n ) | ≤ X ( x ,...,x k ) ∈ η k = | f ( n ) ( x , . . . , x k ) | ≤ X ( x ,...,x k ) ∈ η k = | f ( x , . . . , x k ) | ∈ L ( P ) . Hence, ( F ( n ) ) n ∈ N converges almost surely to F , and the dominated con-vergence theorem implies even convergence in L ( P ) and L ( P ). Moreover, F ( n ) ∈ L ( P ), and every F ( n ) has a Wiener–Itˆo chaos expansion with kernels f ( n ) i that are simple functions since integration over a variable of a simplefunction leads to a simple function. M. REITZNER AND M. SCHULTE
The fact that the kernels of F ( n ) are simple functions brings us in theposition to use Proposition 4.5 to evaluate R ij and ˜ R i for i, j = 1 , . . . , k . Westart by estimating R ii . By (12), we have R ii = Z X E I i − ( f ( n ) i ( s, · )) I i − ( f ( n ) i ( t, · )) dµ ( s, t ) − [( i − k f ( n ) i k ] = X π ∈ Π i − ,i − ,i − ,i − Z X | π | +2 ( f ( n ) i ( s, · ) ⊗ f ( n ) i ( s, · ) ⊗ f ( n ) i ( t, · ) ⊗ f ( n ) i ( t, · )) π ( y , . . . , y | π | ) dµ ( y , . . . , y | π | , s, t ) − [( i − k f ( n ) i k ] . The sum over those partitions of Π i − ,i − ,i − ,i − such that every block con-tains only variables of the first pair or of the second pair of functions leadsexactly to [( i − k f ( n ) i k ] . These partitions cancel out with the minus termand we denote the remaining partitions by ˜Π i − ,i − ,i − ,i − . Hence, R ii = X ˜ π ∈ ˜Π i − ,i − ,i − ,i − Z X | ˜ π | +2 ( f ( n ) i ( s, · ) ⊗ f ( n ) i ( s, · ) ⊗ f ( n ) i ( t, · ) ⊗ f ( n ) i ( t, · )) ˜ π ( y , . . . , y | ˜ π | ) dµ ( y , . . . , y | ˜ π | , s, t ) ≤ X ˜ π ∈ ˜Π i − ,i − ,i − ,i − Z X | ˜ π | +2 | ( f ( n ) i ( s, · ) ⊗ f ( n ) i ( s, · ) ⊗ f ( n ) i ( t, · ) ⊗ f ( n ) i ( t, · )) ˜ π ( y , . . . , y | ˜ π | ) | dµ ( y , . . . , y | ˜ π | , s, t ) . In order to simplify our notation, we include s and t into the partitions byadding two blocks generating s and t to the old partition ˜ π and obtain a newpartition π ∈ Π i,i,i,i . By definition of ˜ π , π has at least one block includingvariables z ( l ) i and z ( l ) i , l ∈ { , } , l ∈ { , } . By construction of π , there arealso blocks including variables of the first two functions and of the last twofunctions. Altogether, this implies π ∈ Π i,i,i,i . Since each ˜ π ∈ ˜Π i − ,i − ,i − ,i − leads to a different π ∈ Π i,i,i,i , we obtain the upper bound R ii ≤ X π ∈ Π i,i,i,i Z X | π | | ( f ( n ) i ⊗ f ( n ) i ⊗ f ( n ) i ⊗ f ( n ) i ) π ( y , . . . , y | π | ) | dµ ( y , . . . , y | π | ) . LT’S FOR POISSON U -STATISTICS In the very same way, we obtain an upper bound for R ij , i = j . By (12), itfollows R ij = Z X E [ I i − ( f ( n ) i ( s, · )) I j − ( f ( n ) j ( s, · )) I i − ( f ( n ) i ( t, · )) × I j − ( f ( n ) j ( t, · ))] dµ ( s, t )= X ˜ π ∈ Π i − ,j − ,i − ,j − Z X | ˜ π | +2 ( f ( n ) i ( s, · ) ⊗ f ( n ) j ( s, · ) ⊗ f ( n ) i ( t, · ) ⊗ f ( n ) j ( t, · )) ˜ π ( y , . . . , y | ˜ π | ) dµ ( y , . . . , y | ˜ π | , s, t ) ≤ X π ∈ Π i,j,i,j Z X | π | | ( f ( n ) i ⊗ f ( n ) j ⊗ f ( n ) i ⊗ f ( n ) j ) π ( y , . . . , y | π | ) | dµ ( y , . . . , y | π | ) . Since i = j , there exist no ˜ π ∈ Π i − ,j − ,i − ,j − with blocks including eithervariables of the first two or last two functions. Hence, we obtain partitions π ∈ Π i,j,i,j by the same construction as for R ii and obtain an upper boundby summing over Π i,j,i,j .The last step is to estimate ˜ R i . Here, we have in a similar way˜ R i = Z X E I i − ( f ( n ) i ( s, · )) dµ ( s )= X ˜ π ∈ Π i − ,i − ,i − ,i − Z X | ˜ π | +1 ( f ( n ) i ( s, · ) ⊗ f ( n ) i ( s, · ) ⊗ f ( n ) i ( s, · ) ⊗ f ( n ) i ( s, · )) ˜ π ( y , . . . , y | ˜ π | ) dµ ( y , . . . , y | ˜ π | , s ) ≤ X π ∈ Π i,i,i,i Z X | π | | ( f ( n ) i ⊗ f ( n ) i ⊗ f ( n ) i ⊗ f ( n ) i ) π ( y , . . . , y | π | ) | dµ ( y , . . . , y | π | ) . In this case, it is immediate that we obtain a partition π ∈ Π i,i,i,i by adding s to a partition ˜ π ∈ Π i − ,i − ,i − ,i − . Thus R ij and ˜ R i are bounded by thesame expressions.Now it remains to estimate the kernels f ( n ) i . From Lemma 3.5, it followsthat | f ( n ) i ( y , . . . , y i ) |≤ (cid:18) ki (cid:19) Z X k − i | f ( n ) ( y , . . . , y i , x , . . . , x k − i ) | dµ ( x , . . . , x k − i ) . M. REITZNER AND M. SCHULTE
We obtain the following expression as an upper bound for R ij and ˜ R i . With M ij ( · ) defined by M ij ( g ) = (cid:18) ki (cid:19) (cid:18) kj (cid:19) × X π ∈ Π i,j,i,j Z X | π | +4 k − i − j | ( g ( · , x (1)1 , . . . , x (1) k − i ) ⊗ g ( · , x (2)1 , . . . , x (2) k − j )(14) ⊗ g ( · , x (3)1 , . . . , x (3) k − i ) ⊗ g ( · , x (4)1 , . . . , x (4) k − j )) π ( y , . . . , y | π | ) | dµ ( x (1)1 , . . . , x (4) k − j , y , . . . , y | π | ) , where π acts on the first i , respectively, j variables of g : X k → R , we have R ij ≤ M ij ( f ( n ) ) and ˜ R i ≤ M ii ( f ( n ) ) for 1 ≤ i, j ≤ k. Since in the definition of M ij every block of a partition π ∈ Π i,j,i,j has atleast two elements, the integration in (14) runs over at most 4 k − i − j variables. For i = j = 1 the only partition in Π i,j,i,j is the partition with oneblock and the integration runs over 4 k − R ij and ˜ R i with Theorem 4.1 yields: Lemma 4.6.
Suppose F ( n ) = P η k = f ( n ) is a U -statistic of order k with f ( n ) ∈ S ( X k ) and N is a standard Gaussian random variable. Then d W (cid:18) F ( n ) − E F ( n ) √ Var F ( n ) , N (cid:19) ≤ k / X ≤ i ≤ j ≤ k q M ij ( f ( n ) )Var F ( n ) . Together with the fact that M ij ( f ( n ) ) ≤ M ij ( f ) since | f ( n ) | ≤ | f | and thetriangle inequality for the Wasserstein distance, we obtain d W (cid:18) F − E F √ Var
F , N (cid:19) ≤ d W (cid:18) F − E F √ Var
F , F ( n ) − E F ( n ) √ Var F ( n ) (cid:19) + d W (cid:18) F ( n ) − E F ( n ) √ Var F ( n ) , N (cid:19) ≤ d W (cid:18) F − E F √ Var
F , F ( n ) − E F ( n ) √ Var F ( n ) (cid:19) + 2 k / X ≤ i ≤ j ≤ k p M ij ( f )Var F ( n ) . LT’S FOR POISSON U -STATISTICS By the definition of the Wasserstein distance and some straightforward com-putations, it follows that d W (cid:18) F − E F √ Var
F , F ( n ) − E F ( n ) √ Var F ( n ) (cid:19) ≤ E (cid:12)(cid:12)(cid:12)(cid:12) F − E F √ Var F − F ( n ) − E F ( n ) √ Var F ( n ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ E | F ( n ) − F + E F − E F ( n ) |√ Var F ( n ) + (cid:12)(cid:12)(cid:12)(cid:12) E | F − E F |√ Var F − E | F − E F |√ Var F ( n ) (cid:12)(cid:12)(cid:12)(cid:12) . Because of the convergence of ( F ( n ) ) n ∈ N to F in L ( P ) and L ( P ), the right-hand side vanishes for n → ∞ , and we get our main result. Theorem 4.7.
Suppose F is an absolutely convergent U -statistic of or-der k , and N is a standard Gaussian random variable. Then d W (cid:18) F − E F √ Var
F , N (cid:19) ≤ k / X ≤ i ≤ j ≤ k p M ij ( f )Var F with M ij ( f ) defined in (14).
5. Geometric U -statistics. Central limit theorems for geometric U -statistics. In this section, weassume that our intensity measure has the form µ ( · ) = λθ ( · ) with a σ -finitenonatomic measure θ ( · ) and λ ≥
1. We are interested in the behavior of the U -statistic F for λ → ∞ . Definition 5.1. A U -statistic F = P η k = f is a geometric U -statistic ifit satisfies f ( x , . . . , x k ) = g ( λ ) ˜ f ( x , . . . , x k )with g : R → R , and with ˜ f : X k → R not depending on λ .In the case that g = 1 and f = ˜ f , the value of F for a given realization ofthe Poisson point process is only determined by the geometry of the pointsand does not depend on the intensity rate λ of the underlying process. Theterm “geometric” is used to emphasize this behavior. We slightly generalizethis property by allowing our geometric U -statistics to have an intensity re-lated scaling factor since we always consider standardized random variables,where the scaling factor is cancelled out.By ˜ M ij we denote the value of M ij ( ˜ f ), which is defined in (14), for λ = 1.With this notation, the following central limit theorem holds: M. REITZNER AND M. SCHULTE
Theorem 5.2.
Suppose F is an absolutely convergent geometric U -statistic of order k with k f k > and N is a standard Gaussian randomvariable. Then lim λ →∞ Var Fλ k − g ( λ ) = k Z X (cid:18)Z X k − ˜ f ( y, x , . . . , x k − ) dθ ( x , . . . , x k − ) (cid:19) dθ ( y ) | {z } =: ˜ V with ˜ V > , and d W (cid:18) F − E F √ Var
F , N (cid:19) ≤ λ − / (cid:18) k / X ≤ i ≤ j ≤ k q ˜ M ij ˜ V (cid:19) (15) for λ ≥ . The main feature of this theorem is that the term in brackets is inde-pendent of λ , which means that for λ → ∞ the distance to the Gaussiandistribution tends to zero at a rate λ − / . Proof of Theorem 5.2.
Because we are interested in the standardizedvariable ( F − E F ) / √ Var F which is independent of g ( λ ), w.l.o.g. we put g ( λ ) = 1 and ˜ f = f . From formula (6) we inferVar F = k X i =1 λ k − i i ! (cid:18) ki (cid:19) × Z X i (cid:18)Z X k − i f ( y , . . . , y i , x , . . . , x k − i ) dθ ( x , . . . , x k − i ) (cid:19) dθ ( y , . . . , y i ) , which means that the variance is a polynomial of degree 2 k − V λ k − = k f k > F ≥ ˜ V λ k − .As previously mentioned, the integration in M ij ( f ) runs over at most4 k − i − j ≤ k − i, j ) = (1 ,
1) and 4 k − i, j ) =(1 , M ij ( f ) ≤ ˜ M ij λ k − for λ ≥
1. Hence, Theorem 4.7 leads directly to (15). (cid:3)
The assumption k f k > Example.
Let η be a Poisson process on [ − ,
1] with intensity measurethe Lebesgue measure times intensity λ >
0. We define the U -statistic F = LT’S FOR POISSON U -STATISTICS P ( x ,x ) ∈ η = f ( x , x ) with f ( x , x ) = (cid:26) , x x ≥ , − , x x < f ( y ) = 0. It is possible to rewrite F as F = L ( L −
1) + R ( R − − LR where L and R are the number of points in [ − ,
0] and[0 , λ , and the thirdmoment of F is 64 λ . Thus the third moment of ( F − E F ) / √ Var F tendsto a constant and hence is too large for convergence of F to a Gaussiandistribution. By a technical computation of all moments, using the productformula for multiple Wiener–Itˆo integrals, for example, and the method ofmoments, it can be shown that √ F − E F ) / √ Var F follows a centeredchi-square distribution with one degree of freedom as λ → ∞ .In the special case µ ( X ) = λθ ( X ) < ∞ , it is possible to approximate thePoisson point process η by a binomial point process, that consists of a fixednumber of independently distributed points with the probability measure θ ( · ) /θ ( X ). If we sum over k -tuples of distinct points of the binomial pointprocess instead of a Poisson point process, we obtain a classical U -statistic.This well-known class of random variables satisfies a similar central limittheorem as above with a rate of convergence; see [7, 11, 16, 20]. Althoughboth results are similar, it seems to be difficult to prove one result by theother, especially with keeping rates of convergence.For classical U -statistics the so-called Hoeffding decomposition which isclosely related to the Wiener–Itˆo chaos expansion plays a crucial role. In therecent paper by Lachi´eze-Rey and Peccati [18], this decomposition is appliedto U -statistics of Poisson point processes which yields a representation sim-ilar to the Wiener–Itˆo chaos expansion. Combining this with the result ofDynkin and Mandelbaum [6], the authors derive our Theorem 5.2 for thecase µ ( X ) < ∞ (without rates of convergence). They also prove noncentrallimit theorems for the case that some of the first kernels of the chaos expan-sion of a U -statistic vanish, which allows one to deal with situations as inthe previous example.In Sections 5.2 and 5.3, we apply Theorem 5.2 to problems from stochasticgeometry. In the recent paper [5] the underlying result from [27] is used toderive a central limit theorem for the number random simplices on a torus.This problem exactly fits in the framework of geometric U -statistics, andsome of the results can also be obtained by using Theorem 5.2.5.2. Central limit theorems for Poisson hyperplanes.
We use Theorem 5.2to establish central limit theorems for Poisson hyperplane processes. Let η be a Poisson process on the space H of all hyperplanes in R d with an inten-sity measure of the form µ ( · ) = λθ ( · ) with λ ∈ R + and a σ -finite nonatomic M. REITZNER AND M. SCHULTE measure θ . The Poisson hyperplane process is only observed in a compactconvex window W ⊂ R d with interior points. Thus, we can view η as a Pois-son process on the set [ W ] defined by[ W ] = { h ∈ H : h ∩ W = ∅ } . Given the hyperplane process η , we investigate the ( d − k )-flats in W which occur as the intersection of k hyperplanes of η . In particular, we areinterested in the sum of their i th intrinsic volumes given byΦ ki ( W ) = 1 k ! X ( h ,...,h k ) ∈ η k = V i ( h ∩ · · · ∩ h k ∩ W )for i = 0 , . . . , d − k and k = 1 , . . . , d . For the definition of the i th intrinsicvolume V i ( · ) we refer to [31]. We remark that V ( K ) is the Euler charac-teristic of the set K , and that V n ( K ) of an n -dimensional convex set K isthe Lebesgue measure Λ n ( K ). Thus Φ k is the number of ( d − k )-flats in W ,and Φ kd − k is their ( d − k )-volume. To ensure that the expectations of theserandom variables are neither 0 nor infinite, we assume that: • < θ ([ W ]) < ∞ ; • ≤ k ≤ d independent random hyperplanes on [ W ] with probability mea-sure θ ( · ) /θ ([ W ]) intersect in a ( d − k )-flat almost surely and their inter-section flat hits the interior of W with positive probability.For example, these conditions are satisfied if the hyperplane process is sta-tionary and the directional distribution is not concentrated on a great sub-sphere.The fact that the summands in the definition of Φ ki are bounded and havea bounded support makes sure that the fourth moments in M ij ( · ) are finite,and we can apply Theorem 5.2: Theorem 5.3.
Let N be a standard Gaussian random variable. Thenconstants c Φ ( W, k, i ) exist such that d W (cid:18) Φ ki ( W ) − E Φ ki ( W ) q Var Φ ki ( W ) , N (cid:19) ≤ c Φ ( W, k, i ) λ − / for λ ≥ , i = 0 , . . . , d − k and k = 1 , . . . , d . Furthermore, the asymptotic variances are given bylim λ →∞ Var Φ ki ( W ) λ k − = 1( k − LT’S FOR POISSON U -STATISTICS × Z [ W ] (cid:18)Z [ W ] k − V i ( h ∩ h ∩ · · · ∩ h k − ∩ W ) dθ ( h , . . . , h k − ) (cid:19) dθ ( h ) . Similar results have first been derived by Paroux [26], and by Heinrich[9] and Heinrich, Schmidt and Schmidt [10] using Hoeffding’s decompositionof classical U -statistics. Schulte and Th¨ale [33] used the Wiener–Itˆo chaosexpansion to compute the moments and cumulants and to formulate centrallimit theorems for the surface area of Poisson hyperplanes in an increasingwindow. In their recent paper [34] this approach is further refined to obtainpoint process convergence for the intrinsic volumes of the intersection processof Poisson k -flats in the unit ball.5.3. Convex hulls of random points.
In the following, we assume thatthe Poisson point process η has an intensity-measure of the form µ ( · ) = λ Λ d ( · ∩ K ), λ ≥
1, where Λ d is Lebesgue measure, and K ⊂ R d a compactconvex set with Λ d ( K ) = 1. If we integrate with respect to Λ d , we omit themeasure in our notation.We consider the following functional related to Sylvester’s problem: H = X ( x ,...,x k ) ∈ η k = h ( x , . . . , x k )with h ( x , . . . , x k ) = ( x , . . . , x k are vertices of conv( x , . . . , x k )) , which counts the number of k -tuples of the process such that every point is avertex of the convex hull, that is, the number of k -tuples in convex position.The expected value of H is then given by E H = λ k P ( X , . . . , X k are vertices of conv( X , . . . , X k )) = λ k p ( k ) ( K ) , where X , . . . , X k are independent random points chosen according to theuniform distribution on K .The question to determine the probability p ( k ) ( K ) that k random points ina convex set K are in convex position has a long history; see, for example, themore recent developments by B´ar´any [1, 2] and Buchta [3]. In our setting, thefunction H is an estimator for the probability p ( k ) ( K ), and we are interestedin distributional properties of this estimator.The asymptotic behavior of Var H is determined by˜ H = lim λ →∞ Var Hλ k − = k Z K (cid:18)Z K k − h ( y, x , . . . , x k − ) dx · · · dx k − (cid:19) dy. M. REITZNER AND M. SCHULTE
By the Cauchy–Schwarz inequality, because Λ d ( K ) = 1 and h = h , we ob-tain ˜ H ≤ k Z K Z K k − h ( y, x , . . . , x k − ) dx · · · dx k − dy = k p ( k ) ( K )and k p ( k ) ( K ) = k (cid:18)Z K k h ( x , . . . , x k ) dx · · · dx k (cid:19) ≤ k Z K (cid:18)Z K k − h ( x , x , . . . , x k ) dx · · · dx k (cid:19) dx = ˜ H. Together with Theorem 5.2, we immediately get the following result showingthat the estimator H is asymptotically Gaussian: Theorem 5.4.
Let N be a standard Gaussian random variable. Thenthere exists a constant C such that d W (cid:18) H − E H √ Var
H , N (cid:19) ≤ Cλ − / . Furthermore
Var H = λ k − ˜ H (1 + O ( λ − )) as λ → ∞ with k p ( k ) ( K ) ≤ ˜ H ≤ k p ( k ) ( K ) .
6. Local U -statistics. Central limit theorems for local U -statistics. For a geometric U -statistic the function f is [up to the scaling factor g ( λ )] independent of λ .Now we allow that f is influenced by λ in a more intricate way, but weassume that a k -tupel of points is only in the support of f if the points areclose together.From now on, let X be a metric space, and denote by B ( y, r ) the ballwith center y and radius r . Again, we assume that the intensity measure µ has the form µ ( · ) = λθ ( · ) with λ ≥ σ -finite nonatomic measure θ ( · )on X . We denote the diameter of A ⊂ X by diam( A ). Definition 6.1. A U -statistic F = P η k = f is a local U -statistic if itsatisfies f ( x , . . . , x k ) = 0 if diam( { x , . . . , x k } ) > δ. (16)Note that in general δ may depend on λ . We denote the L -norm on X i with respect to the measure θ ( · ) by k · k θ . Now we can rephrase Theorem4.7 for local U -statistics as follows: LT’S FOR POISSON U -STATISTICS Theorem 6.2.
Suppose F is an absolutely convergent local U -statisticof order k with k f k > , and N is a standard Gaussian random variable.Putting ˜ V = k f k /λ k − and b ( δ ) = max y ∈ X µ ( B ( y, δ )) < ∞ , we have d W (cid:18) F − E F √ Var
F , N (cid:19) ≤ c k λ − k/ max { , b ( δ ) (3 k − / } k f k θ ˜ V with a constant c k ∈ R only depending on k . Proof.
Formula (6) yields Var F ≥ k f k = λ k − ˜ V . The estimate for M ij runs as follows. Since π ∈ Π i,j,i,j , and condition (16) forces all argumentsof f to be close, we can rewrite M ij ( f ) as M ij ( f ) = (cid:18) ki (cid:19) (cid:18) kj (cid:19) × X π ∈ Π i,j,i,j Z X | π | +4 k − i − j | ( f ( · , x (1)1 , . . . , x (1) k − i ) ⊗ f ( · , x (2)1 , . . . , x (2) k − j ) ⊗ f ( · , x (3)1 , . . . , x (3) k − i ) ⊗ f ( · , x (4)1 , . . . , x (4) k − j )) π ( y , . . . , y | π | ) |× (diam( { x (1)1 , . . . , x (4) k − j , y , . . . , y | π | } ) ≤ δ ) dµ ( x (1)1 , . . . , x (4) k − j , y , . . . , y | π | ) . By H¨older’s inequality, we obtain M ij ( f ) ≤ c ij X π ∈ Π i,j,i,j Z X | π | +4 k − i − j f ( z , . . . , z k ) × (diam( { z , . . . , z | π | +4 k − i − j } ) ≤ δ ) dµ ( z , . . . , z | π | +4 k − i − j ) ≤ c ij k f k X π ∈ Π i,j,i,j b ( δ ) | π | +3 k − i − j = c ij λ k k f k θ X π ∈ Π i,j,i,j b ( δ ) | π | +3 k − i − j with a constant c ij ∈ R depending on i, j, k . One should keep in mind thatmax( i, j ) ≤ | π | ≤ i + j for all π ∈ Π i,j,i,j and that the only partition π ∈ M. REITZNER AND M. SCHULTE Π , , , satisfies | π | = 1. This leads to | π | − i − j ≤ − k / X ≤ i ≤ j ≤ k p M ij ( f )Var F ≤ c ′ k X ≤ i ≤ j ≤ k q λ k k f k θ P π ∈ Π i,j,i,j b ( δ ) | π | +3 k − i − j λ k − ˜ V ≤ c ′ k λ − k/ k f k θ ˜ V X ≤ i ≤ j ≤ k s X π ∈ Π i,j,i,j b ( δ ) | π | +3 k − i − j ≤ c k λ − k/ k f k θ ˜ V max { , b ( δ ) (3 k − / } with constants c ′ k , c k ∈ R only depending on k . Combining this estimate withTheorem 4.1 gives the claimed result. (cid:3) The proof rests essentially upon the fact that F is a local U -statisticsince this allows us to rewrite M ij ( f ) such that every function depends onall variables and to split these functions using H¨older’s inequality.6.2. A central limit theorem for the total edge length of a random geomet-ric graph.
We apply the results of the previous subsection to a problemfrom random graph theory. We construct a random graph in the followingway. Let η be a Poisson process in X = R d with an intensity measure of theform µ ( · ) = λ Λ d ( · ∩ W )with λ ≥
1, the d -dimensional Lebesgue-measure Λ d ( · ) and a compact win-dow W ⊂ R d of volume Λ d ( W ) = 1 containing the origin in its interior. Weregard η as a set of points in W . As in (16) we connect two points x, y ∈ η by an edge if k x − y k ≤ δ = δ ( λ ) . The resulting graph G ( P λ , δ ) is a random geometric graph, sometimes calleda Gilbert graph or an interval graph (for d = 1) and a disk graph (for d = 2).For graph-theoretical properties of G ( P λ , δ ) we refer to [30] and to the morerecent developments [8, 17, 21, 23]. For our central limit theorem we take λ → ∞ and assume that δ is small enough to ensure that \ x ∈ B (0 ,δ ) ( W + x ) ⊃ W. We are interested in the total edge length L ( η ) of G ( P λ , δ ) in the window W , which is given by L ( η ) = 12 X ( x,y ) ∈ η = g ( x − y ) ( k x − y k ≤ δ ) . LT’S FOR POISSON U -STATISTICS Here g : B (0 , δ ) → R is some kind of measure of the length of the edge ( x, y ).We assume g ∈ L ( B (0 , δ )) which implies that L is absolutely convergent.The following lemma is immediate from Lemma 3.5. Lemma 6.3. L ( η ) has a Wiener–Itˆo chaos expansion with kernels f ( y ) = λ Z B (0 ,δ ) g ( x ) ( y + x ∈ W ) dx, y ∈ W and f ( x, y ) = g ( x − y ) ( k x − y k ≤ δ ) , x, y ∈ W. For the length of this random graph, we obtain the following central limittheorem:
Theorem 6.4.
Assume g ∈ L ( B (0 , δ )) with R B (0 ,δ ) g ( x ) dx = 0 , and let N be a standard Gaussian random variable. Then there is a constant c d onlydepending on the dimension d such that d W (cid:18) F − E F √ Var
F , N (cid:19) ≤ c d λ − max { , b ( δ ) / } ( R B (0 ,δ ) g ( x ) dx ) / ( R B (0 ,δ ) g ( x ) dx ) . Proof.
We compute the bound from Theorem 6.2. Lemma 6.3 yields˜ V = k f k λ = Z W (cid:18)Z B (0 ,δ ) g ( x ) ( y + x ∈ W ) dx (cid:19) dy ≥ Z (1 / W (cid:18)Z B (0 ,δ ) g ( x ) dx (cid:19) dy = 2 − d (cid:18)Z B (0 ,δ ) g ( x ) dx (cid:19) and k f k θ = 116 Z W Z B (0 ,δ ) g ( x ) ( y + x ∈ W ) dx dy ≤ Z B (0 ,δ ) g ( x ) dx. (cid:3) As an example we consider the particular case g = 1, where L ( η ) re-duces to the number of edges of the graph. Then the expectation is of order λ δ d . Lemma 6.3 and Theorem 6.4 tell us that the variance is of ordermax { λ δ d , λ δ d } and that d W (cid:18) L − E L √ Var
L , N (cid:19) ≤ ˜ c d λ − δ − d/ max { , λ / δ d/ } with a constant ˜ c d ∈ R only depending on d . The right-hand side tends tozero if λ / Λ d ( B (0 , δ )) → ∞ as λ → ∞ . In the maybe most natural case when λ Λ d ( B (0 , δ )) stays constant we have an order λ − / of convergence to the M. REITZNER AND M. SCHULTE
Gaussian distribution. A central limit theorem without rate of convergenceis a special case of Theorem 3.9 in [30].Similar results to Lemma 6.3 and Theorem 6.4 can be obtained if theintensity measure is of the form dµ ( x ) = λf ( x ) d Λ d ( x ) with λ ∈ R + and adensity function f ( x ). Acknowledgments.
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Institut f¨ur MathematikUniversit¨at Osnabr¨uck49069 Osnabr¨uckGermanyE-mail: [email protected]