Normal approximation of geometric Poisson functionals

Transcription

1 Institut für Stochastik Karlsruher Institut für Technologie Normal approximation of geometric Poisson functionals (Karlsruhe) joint work with Daniel Hug, Giovanni Peccati, Matthias Schulte presented at the Simons Workshop on Stochastic Geometry and Point Processes UT Austin May 5-8, 2014

2 1. Chaos expansion of Poisson functionals Setting η is a Poisson process on some measurable space (X, X ) with intensity measure λ. This is a random element in the space N of all integer-valued σ-finite measures on X, equipped with the usual σ-field (and distribution Π λ ) with the following two properties The random variables η(b 1 ),..., η(b m ) are stochastically independent whenever B 1,..., B m are measurable and pairwise disjoint. P(η(B) = k) = λ(b)k k! exp[ λ(b)], k N 0, B X, where k e := 0 for all k N 0.

3 Definition (Difference operator) For a measurable function f : N R and x X we define a function D x f : N R by D x f (µ) := f (µ + δ x ) f (µ). For x 1,..., x n X we define D n x 1,...,x n f : N R inductively by where D 1 := D and D 0 f = f. Lemma For any f L 2 (P η ) defines a function T n f L 2 s(λ n ). D n x 1,...,x n f := D 1 x 1 D n 1 x 2,...,x n f, T n f (x 1,..., x n ) := ED n x 1,...,x n f (η),

4 Definition Let n N and g L 2 (λ n ). Then I n (g) denotes the multiple Wiener-Itô integral of g w.r.t. the compensated Poisson process η λ. For c R let I 0 (c) := c. These integrals have the properties EI n (g)i n (h) = n! g, h n, n N 0, EI m (g)i n (h) = 0, m n. Here g(x 1,..., x n ) := 1 g(x n! π(1),..., x π(n) ) π Σ n denotes the symmetrization of g.

5 Definition Let L 2 η denote the space of all random variables F L 2 (P) such that F = f (η) P-almost surely, for some measurable function (representative) f : N R. Theorem (Wiener 38; Itô 56; Y. Ito 88; L. and Penrose 11) For any F L 2 η there are uniquely determined f n L 2 s(λ n ) such that P-a.s. F = EF + I n (f n ). n=1 Moreover, we have f n = 1 n! T nf, where f is a representative of F.

6 2. Malliavin operators Definition Let F L 2 η have representative f. Define D x F := D x f (η) for x X, and, more generally D n x 1,...,x n F := D n x 1,...,x n f (η) for any n N and x 1,..., x n X. The mapping (ω, x 1,..., x n ) D n x 1,...,x n F(ω) is denoted by D n F (or by DF in the case n = 1).

7 Theorem (Y. Ito 88; Nualart and Vives 90; L. and Penrose 11) Suppose F = EF + n=1 I n(f n ) L 2 η. Then DF L 2 (P λ) iff F is in dom D (the domain of the Malliavin difference operator), that is nn! fn 2 dλ n <. n=1 In this case we have P-a.s. and for λ-a.e. x X that D x F = ni n 1 (f n (x, )). n=1

8 Definition Let H L 2 η(p λ). Define h 0 (x) := EH(x) and h n (x, x 1,..., x n ) := 1 n! EDn x 1,...,x n H(x) and assume that H is in dom δ (the domain of the operator δ), that is (n + 1)! n=0 h 2 n dλ n+1 <. Then the Kabanov-Skorohod integral δ(h) of H is given by δ(h) := I n+1 (h n ). n=0

9 Theorem (Nualart and Vives 90) Let F dom D and H dom δ. Then E (D x F)H(x) λ(dx) = EFδ(H). Theorem (Picard 96; L. and Penrose 11) Let H L 1 η(p λ) L 2 η(p λ) be in the domain of δ and have representative h. Then δ(h) = h(η δ x, x) η(dx) h(η, x) λ(dx) P-a.s.

10 Definition The domain dom L of the Ornstein-Uhlenbeck generator L is given by all F L 2 η satisfying n 2 n! f n 2 n <. n=1 For F dom L one defines LF := ni n (f n ). n=1 The (pseudo) inverse L 1 of L is given by L 1 F := n=1 1 n I n(f n ).

11 3. Normal approximation: General results Definition The Wasserstein distance between the laws of two random variables Y 1, Y 2 is defined as d W (Y 1, Y 2 ) = sup Eh(Y 1 ) Eh(Y 2 ). h Lip(1) Theorem (Peccati, Solé, Taqqu and Utzet 10) Let F dom D such that EF = 0 and N be standard normal. Then d W (F, N) E 1 (D x F)( D x L 1 F ) λ(dx) + E (D x F) 2 D x L 1 F λ(dx).

12 Definition The Kolmogorov distance between the laws of two random variables Y 1, Y 2 is defined by d K (Y 1, Y 2 ) = sup P(Y 1 x) P(Y 2 x). x R

13 Theorem (Schulte 12) For F dom D and N standard normal d K (F, N) where [ c(f) = E [ + E [ ( E 1 ) 2 ] 1/2 (D x F)( D x L 1 F)λ(dx) [ ] 1/2 + 2c(F) E (D x F) 2 (D x L 1 F) 2 λ(dx) + sup E D x 1{F > t}(d x F) D x L 1 F λ(dx), t R ] 1/2 (D x F) 4 λ(dx) 1/4 ( ) (D x F) 2 (D y F) 2 λ 2 (d(x, y))] (EF 4 ) 1/4 + 1.

14 Theorem (Hug, L. and Schulte 13) Let F L 2 η have the chaos expansion F = EF + I n (f n ) n=1 and assume that Var F > 0. Assume that there are a > 0 and b 1 such that (f m f m f n f n ) σ dλ σ a bm+n (m!) 2 (n!) 2 for all σ Π mn. Let N be a standard Gaussian random variable. Then, under an additional integrability assumption, ( ) F EF d W, N Var F n=1 n 17/2 b n n/14! a Var F.

15 Theorem (L., Peccati and Schulte 14) Let F dom D be such that EF = 0 and Var F = 1, and let N be standard Gaussian. Then, where d W (F, N) γ 1 + γ 2 + γ 3, γ1 2 := 16 [E(Dx1 F ) 2 (D x2 F) 2] 1/2[ E(D 2 x1,x 3 F) 2 (Dx 2 2,x 3 F) 2] 1/2 dλ 3, γ2 2 := E(Dx 2 1,x 3 F) 2 (Dx 2 2,x 3 F) 2 dλ 3, γ 3 := E D x F 3 λ(dx).

16 Theorem (L., Peccati and Schulte 14) For F, G dom D with EF = EG = 0, we have ( E Cov(F, G) ) 2 (D x F)( D x L 1 G) λ(dx) [E(D 2 x 1,x 3 F) 2 (D 2 x 2,x 3 F) 2] 1/2[ E(Dx1 G) 2 (D x2 G) 2] 1/2 dλ 3 [E(Dx1 F)2 (D x2 F)2] 1/2[ E(D 2 x 1,x 3 G) 2 (D 2 x 2,x 3 G) 2] 1/2 dλ 3 [E(D 2 x 1,x 3 F) 2 (D 2 x 2,x 3 F) 2] 1/2[ E(D 2 x1,x 3 G) 2 (D 2 x 2,x 3 G) 2] 1/2 dλ 3.

17 4. The Boolean model

18 4. The Boolean model

19 4. The Boolean model

20 Setting η is a Poisson process on K d (the space of convex bodies) with intensity measure Λ of the translation invariant form Λ( ) = γ 1{K + x } dx Q(dK ), where γ > 0, and Q is a probability measure on K d with Q({ }) = 0 and V d (K + C)Q(dK ) <, C R d compact.

21 Definition The Boolean model (based on η) is the random closed set Z := K η K. Remark The Boolean model is stationary, that is Z + x d = Z, x R d. Remark The intersection Z W of the Boolean model Z with a convex set W K d belongs to the convex ring R d, that is, Z W is a finite union of convex bodies.

22 4.1 Mean values Theorem Let ψ : R d R be measurable, additive, translation invariant and locally bounded. Let W K d with V d (W ) > 0. Then the limit Eψ(Z rw ) δ ψ := lim r V d (rw ) exists and is given by δ ψ = E[ψ(Z [0, 1] d ) ψ(z + [0, 1] d )], where + [0, 1] d is the upper right boundary of the unit cube [0, 1] d. Moreover, due to ergodicity, there is a pathwise version of this convergence.

23 Example The intrinsic volumes V 0,..., V d have all properties required by the theorem. For K K d, they are defined by the Steiner formula V d (K + rb d ) = d r j κ j V d j (K ), r 0, j=0 where κ j is the volume of the Euclidean unit ball in R j. The intrinsic volumes can be additively extended to the convex ring. Definition Let Z 0 be typical grain, that is, a random closed set with distribution Q and define v i := EV i (Z 0 ) = V i (K )Q(dK ), i = 0,..., d,

24 Example The volume fraction of Z is defined by p := EV d (Z [0, 1] d ) = P(0 Z ) and given by the formula p = 1 exp[ γv d ]. Note that Eλ d (Z B) = pλ d (B), B B(R d ), that is δ Vd = p.

25 Example For any W K d, EV d 1 (Z W ) = V d (W )(1 p)γv d 1 + V d 1 (W )p. Therefore the surface density δ Vd 1 of Z is given by the formula δ Vd 1 = (1 p)γv d 1. Definition The Boolean model is called isotropic if Q is invariant under rotations or, equivalently, if for all rotations ρ : R d R d. Z d = ρz

26 Theorem (Miles 76, Davy 78) Assume that Z is isotropic and let j {0,..., d}. Then EV j (Z W ) V j (W ) = (1 p) d V k (W )P j,k (γv j,..., γv d 1 ) for any W K d, where the polynomials P j,k are defined below. In particular k=j δ Vj = (1 p)p j,d (γv j,..., γv d 1 ). Remark The first formula can be extended to more general additive functionals.

27 Definition For j {0,..., d 1} and k {j,..., d} define a polynomial P j,k on R d j of degree k j by P j,k (t j,..., t d 1 ) := 1{k = j} + c k j k j s=1 ( 1) s s! d 1 m 1,...,m s=j m m s=sd+j k s i=1 c m i d t m i, where c k j := k!κ k j!κ j.

28 4.2 Covariance structure Definition For p 1 the integrability assumption IA(p) holds if EV i (Z 0 ) p <, i = 0,..., d, Assumption IA(2) is assumed to hold throughout the rest of this section.

29 Definition Let C W (x) := V d (W (W + x)), x R d, be the covariogram of W K d and C d (x) := EV d (Z 0 (Z 0 + x)), x R d, the mean covariogram of the typical grain. Theorem We have P(0 Z, x Z ) p 2 = (1 p) 2( e γc d (x) 1 ), x R d, Var(V d (Z W )) = (1 p) 2 C W (x) ( e γc d (x) 1 ) dx, W K d.

30 Definition Let W K d satisfy V d (W ) > 0. The asymptotic covariances of the intrinsic volumes are defined by Cov(V i (Z rw ), V j (Z rw )) σ i,j := lim, i, j = 0,..., d. r V d (rw ) Example As it is well-known that C lim W (x) r(w ) V d (W ) = 1, where r(w ) denotes the inradius of W, it follows that σ d,d = (1 p) 2 (e γc d (x) 1 ) dx.

31 Theorem (Hug, L. and Schulte 13) The asymptotic covariances σ i,j exist. Moreover, there is a constant c > 0 depending only on the dimension and Λ, such that Cov(V i (Z rw ), V j (Z rw )) V d (rw ) σ i,j Moreover, this rate of convergence is optimal. c r(w ).

32 Main ideas of the Proof: The Fock space representation of Poisson functionals (cf. L. and Penrose 11) gives for any F, G L 2 η, that Cov(F, G) 1 = n! n=1 (ED n K 1,...,K n F)(ED n K 1,...,K n G) Λ n (d(k 1,..., K n )). For an additive functional ψ : R d R define f ψ,w (µ) := ψ(z (µ) W ), where Z (µ) is the union of all grains charged by the counting measure µ. Then D n K 1,...,K n f ψ,w (µ) = ( 1) n (ψ(z (µ) K 1... K n W ) ψ(k 1... K n W )).

33 One can prove that for all n N and K 1,..., K n K d ED n K 1,...,K n f ψ,w (η) β(ψ) d V i (K 1... K n W ), where the constant β(ψ) does only depend on ψ, Λ and the dimension. Applying the (new) integral geometric inequalities i=0 d d V k (A (K + x))dx β 1 V i (A) V r (K ), A K d, i=0 r=k where β 1 depends only on the dimension, and d d V k (A K 1... K n ) Λ n (d(k 1,..., K n )) α n V k (A), k=0 k=0 where α := γ(d + 1)β 1 d i=0 EV i(z 0 ) allows to use dominated convergence to derive the result.

34 4.3 Normal approximation Theorem (Hug, L. and Schulte 13) Assume that IA(4) holds. Let W K d with r(w ) 1, i {0,..., d}, and N be a centred Gaussian random variable. Then, for all W K d with sufficiently large inradius, d W ( Var(Vi (Z W )) 1 (V i (Z W ) EV i (Z W )), N ) cv d (W ) 1/2, where the constant c > 0 depends only on Λ and the dimension. If only IA(2) holds, then we still have convergence in distribution.

35 Main ideas of the Proof: Let f n (K 1,..., K n ) := ( 1)n ( EVi (Z K 1 K n W ) V i (K 1 K n W ) ) n! be the kernels of the chaos expansion of V i (Z W ). Bound (f m f m f n f n ) σ dλ σ using the previous integral geometric inequalities and apply one of the previous theorems.

36 Remark The above result remains true for any additive, locally bounded and measurable (but not necessarily translation invariant) functional ψ, provided that the variance of ψ(z W )/V d (W ) 1/2 does not degenerate for large W. Theorem (Hug, L. and Schulte 13) If the typical grain Z 0 has nonempty interior with positive probability, then the covariance matrix (σ i,j ) is positive definite.

37 5. Poisson-Voronoi tessellation

38 5. Poisson-Voronoi tessellation

39 Setting η is a stationary Poisson process on R d with unit intensity. Definition The Poisson-Voronoi tessellation is the collection of all cells C(x, η) = {y R d : x y z y, z η}, x η. For k {0,..., d} let X k denote the system of all k-faces of the tessellation.

40 Theorem (Avram and Bertsimas 93, Penrose and Yukich 05, L., Peccati and Schulte 14) Fix W K d and let V (k,i) r := G X V i (G rw ), r > 0, where k {0,..., d}, i {0,..., min{k, d 1}}. There are constants c k,i, such that ( (k,i) V d r W EV (k,i) r Var V (k,i) r ), N c k,i r d/2, r 1. A similar result holds for the Kolmogorov distance.

41 Main ideas of the Proof: Use the stabilizing properties of the Poisson Voronoi tessellation to show that sup E D x V (k,i) 5 r + sup sup E D x,yv 2 (k,i) 5 r < x x,y sup r 1 r 1 and, for q := 1/20, P(D x V (k,i) r 0) q dx cr d, rw ( 2 P(Dx,yV 2 (k,i) r 0) dx) q dy cr d. rw rw

42 Show with other methods that lim inf r r d Var V (k,i) r > 0. Combine this with (a consequence of) the second order Poincaré inequality to conclude the result.

43 6. References Avram, F. and Bertsimas, D. (1993). On central limit theorems in geometrical probability. Ann. Appl. Probab. 3, Chatterjee, S. (2009). Fluctuations of eigenvalues and second order Poincaré inequalities. Probab. Theory Related Fields 143, Hug, D., Last, G. and Schulte, M. (2013). Second order properties and central limit theorems for Boolean models. arxiv: Itô, K. (1956). Spectral type of the shift transformation of differential processes with stationary increments. Trans. Amer. Math. Soc. 81, Ito, Y. (1988). Generalized Poisson functionals. Probab. Th. Rel. Fields 77, 1-28.

44 Kabanov, Y.M. (1975). On extended stochastic integrals. Theory Probab. Appl. 20, Last, G., Peccati, G. and Schulte, M. (2014). Normal approximation on Poisson spaces: Mehler s formula, second order Poincaré inequalities and stabilization. arxiv: Last, G. and Penrose, M.D. (2011). Fock space representation, chaos expansion and covariance inequalities for general Poisson processes. Probab. Th. Rel. Fields 150, Nualart, D. and Vives, J. (1990). Anticipative calculus for the Poisson process based on the Fock space. In: Séminaire de Probabilités XXIV, Lecture Notes in Math., 1426,

45 Peccati, G., Solé, J.L., Taqqu, M.S. and Utzet, F. (2010). Stein s method and normal approximation of Poisson functionals. Ann. Probab. 38, Peccati, G. and Zheng, C. (2010). Multi-dimensional Gaussian fluctuations on the Poisson space. Electron. J. Probab. 15, Penrose, M.D. and Yukich, J.E. (2005). Normal approximation in geometric probability. Barbour, A. and L. Chen (eds.). Stein s method and applications. Singapore University Press. Schulte, M. (2012). Normal approximation of Poisson functionals in Kolmogorov distance. arxiv: Wiener, N. (1938). The homogeneous chaos. Am. J. Math. 60, Wu, L. (2000). A new modified logarithmic Sobolev inequality for Poisson point processes and several applications. Probab. Theory Related Fields 118,