Stein-Malliavin method in the Poisson framework

Transcription

1 Stein-Malliavin method in the Poisson framework Raphaël Lachièze-Rey, Univ. Paris 5 René Descartes Joint work with Giovanni Peccati, University of Luxembourg December 8, 2014 R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

2 1 Stein s method 2 Stochastic Poisson integrals 3 Some geometric problems 4 Malliavin calculus 5 Example: Graph with geometric connections 6 Other limit theorems 7 Semi-group representations R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

3 Limit theorems overview Input alea (Gaussian process, Poisson process, IID variables,...): X Functional:F F px q (regular., local dep., finite order interactions Ñ finite expansion, decreasing expansion) Target Law (Gaussian, Poisson, Gamma,...): U Tools on the target law: Stein s method. Tools on the input law: Stoch. analysis, Malliavin calculus,... Distance used:let H be some class of functions. d H pf px q, Uq sup EhpF px qq hpuq hph R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

4 Distances The distance used can be Kolmogorov distance d K pf, Uq sup PpF ď tq PpU ď tq tpr sup Eh t pf q Eh t puq tpr i.e. H th t ; t P Ru, where h t pxq 1 txďtu. More irregular and difficult to handle, more suited to stat. applications. Wasserstein distance d W pf, Uq sup EhpF q EhpUq h 1-Lipschitz more smooth, H 1 Lipschitz functions R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

5 Stein s method Let N be a standard Gaussian. Then ENf pnq f 1 pnq 0 for f smooth. If reciprocally some variable V satisfies then V is Gaussian. EVf pv q f 1 pv q 0, The idea of Stein s method (with a Gaussian target law) is that if F «N, then EFf pf q f 1 pf q «0 R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

6 Stein s method The trick is to write that for each h P H, there is f h such that hpxq EhpNq f 1 h pxq xf hpxq (1t order ODE) so that replacing x with F and taking the expectation yields EhpF q hpnq ď sup Efh 1 pf q Ff hpf q. h The whole point is that we can choose f h such that }f h } 8, }f h } 1 8 ď c and For Wasserstein distance:, }f 2 h } 8 ď c 1 For Kolmogorov distance f 2 h pxq ď c2 ` x. We have for some adapted distance dpf, Nq ď sup Ef 1 pf q Ff pf q (1) f for f in a class depending on the distance. R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

7 Another basic idea The point is therefore to bound for f with bounded derivatives. Ef 1 pf q Ff pf q An idea that works is to look for a variable T such that ET 1, varpt q is small, EFf pf q ETf 1 pf q because then ˇ ˇEf 1 pf q Ff pf qˇˇ ď ˇˇEf 1 pf qpt 1qˇˇ ` E ˇˇFf pf q Tf 1 pf qˇˇ ď ce T 1 ` ˇˇEFf pf q Tf 1 pf qˇˇ R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

8 How to choose T? Malliavin calculus: T xdv, DL 1 F y 2 Gaussian framework: ETf 1 pf q Ff pf q 0! Poisson framework: Need some kind of chain rule to bound ETf 1 pf q Ff pf q Zero-bias transform: Build F such that for f smooth EFf pf q Ef 1 pf q. Chatterjee (binomial framework). Explicit construction of a variable T such that Ef 1 pf q «Erf 1 pf qept F qs, i.e. in some sense EpT F q «df df, and therefore ErFf pf q Tf 1 pf qs is small R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

9 1 Stein s method 2 Stochastic Poisson integrals 3 Some geometric problems 4 Malliavin calculus 5 Example: Graph with geometric connections 6 Other limit theorems 7 Semi-group representations R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

10 Poisson random measures E underlying Polish space (typically E Ă R d ) µ Radon measure on E X Poisson random measure with intensity µ; noted X Ppµq,such that If A, B Ă E, A X B H, then X paq, X pbq are independent X paq is Poisson with parameter µpaq Since a Poisson measure with parameter n is much concentrated in rn{2, 2ns, X is quite close to n IID points X 1,..., X n with n µpeq. If µ has no atoms, then a.s. X is a finite sum of Diracs in distinct points, and we consider X as a locally finite set X tx n ; n ě 1u We introduce the compensated Poisson measure X X µ. For simplicity we assume that µ has no atoms and X is a set R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

11 Stochastic integrals For f P L 2 pe k ; µ k q symmetric vanishing on the diagonal, one can define the centred random variable ż I k pf k q fdx k. For f k non-vanishing on the diagonal, we have I k pf k q : I k pf k q where f k is zero on the diagonal and equal to f k elsewhere. The k-th Gaussian chaos is the class of all such integrals, and they satisfy, for m, k ě 0, f P L 2 s pe k q, g P L 2 s pe m q EI k pf qi m pgq k!1 m k xf, gy R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

12 U-statistics More likely to appear in applications are the Poisson U-statistics: ż ÿ U k pf ; X q fdx k f px 1,..., x k q px 1,...,x k qpx k whenever this is well defined. It is easy to pass from U-statistics to stochastic integrals and vice versa (assume f 0 on the diagonal): ż fdx k where ż fdppx µq ` µq k f n px 1,..., x n q more convenient notation: f n px n q kÿ n 0 ˆk n ż f n dx k kÿ n 0 ż f px 1,..., x n, x n`1,..., x k qdx 1... dx n. ż f px n, x k n qdµ k n px k n q ˆk I n pf n q n R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

13 Wiener-Ito decomposition Every L 2 -variable F px q admits an infinite decomposition F px q ÿ I n pf n q ně0 It will turn out that the kernels f n P L 2 pe n q can actually be expressed in terms of the n-th order Malliavin derivatives of F. Proof: G tµ ÞÑ e µph 1q ` ` e µph kq u ti k phqu dense in G µpcq lim tñ0 1 t p1 e tµpcq q R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

14 1 Stein s method 2 Stochastic Poisson integrals 3 Some geometric problems 4 Malliavin calculus 5 Example: Graph with geometric connections 6 Other limit theorems 7 Semi-group representations R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

15 Notation Let λ ą 0 l : Lebesgue measure X X λ tx i ; i ě 1u : a homogeneous Poisson process with intensity some loc.finite measure µ λ. X k : Set of k-tuples of distinct points of X. R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

16 Subgraph counting (regimes classification: Penrose 2003) Let α λ ą 0, Let G px, Eq be the graph with edges E tpx i, x j q : }x i x j } ď α λ u (Gilbert graph). Let G be a finite connected graph (e.g. G is a triangle) Then the following random variable is a Poisson U-statistic of order #G: U G px q #toccurrences of G as a subgraph of Gu? µ λ l1 λ 1{d r0,1s d, k #G f px 1,..., x k q 1 tthe subgraph induced on the x 1 i s contains Gu. R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

17 U-statistics in the boolean model ν: Probability measure on the class of compact sets K. To each x i (the germs) is attached an independent random compact set K i (the grain) with distribution νpdk i q. Put X 1 tpx i, K i qu, a marked Poisson point process (product ctrl measure). Two points x i, x j are connected if px i ` K i q X px j ` K j q H. R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

18 Example: Number of intersections in a process of line segments. ν charges isotropically segments centered in 0. Number of segment intersections. U λ : ÿ i j 1 txi is connected to x j u Figure: Zbynek Pawlas R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

19 Sylvester s problem Given a convex body W Ă R d, k ě d ` 1, and X 1,..., X k IID uniform in W, what is p k pw q PpX 1,..., X k are in convex positionq? Explicit formulae for the square, the triangle, or other basic shapes exist. In general, how to estimate p k pw q? If k is large, the accept-reject method is unrealistic. Figure: Are 4 random points in convex position? R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

20 Reitzner & Schulte 2013 proposed the unbiased estimator ÿ ˆp k,λ pw q pλlpw qq k x k : px 1,...,x k qpx k The problem is equivalent if the control measure is l1 λ 1{d W. 1 tx1,...,x k are in convex positionu R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

21 Flat processes Let Γ be a locally finite measure on the set of lines of R d invariant under translations. Let rw s be the class of lines that intersect W Ă R d a convex body. Let X tx i : i ě 1u be a Poisson process with control measure λγ1 rw s. What are the properties of #ptx i X x j X W ; i ă juq? Same question with a process of m-dimensional affine subspaces of R d, with m ą d{2. equivalent problem with control measure l1 λ 1{d rw s R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

22 Stationarity and scaling In our applications, the kernel h λ satisfies h λ px k q h 0 pα λ x k q for some scaling factor α λ ą 0, where h 0 is invariant under translations and permutations: and h 0 px 1 ` y,..., x k ` yq h 0 px 1,..., x k q, y P R d is small far from the origin x k 1 ÞÑ h 0 p0, x k 1 q R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

23 1 Stein s method 2 Stochastic Poisson integrals 3 Some geometric problems 4 Malliavin calculus 5 Example: Graph with geometric connections 6 Other limit theorems 7 Semi-group representations R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

24 Malliavin derivative Technical assumptions all along: Every variable has to be L 2 Every kernel has to be L 2 and symmetric Small perturbation of X in some point x P E: X Ñ X Y txu. D x F px q F px Y txuq F px q. Under the expansion representation, ÿ D x I n pf n q ÿ ni n pf n 1 px, qq ně1 ně0 We also have for ε small, ν a (finite sum of )Dirac measure(s), F px ` ενq «F px q ` εxdf, νy R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

25 Integration by parts For u under the representation u x ÿ I n pf n px, qq ně0 ż ÿ ż n δpu x q f n dpx µq d X ÿ ně0 ně0 ż f n px,... qdpx µq n b px µq Under some assumptions on u, we can write δpuq : I 1 puq 2 ÿ xpx u x ż u x dµpxq. The Malliavin derivative is such that EF δpuq ExDF, uy. R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

26 Higher order derivatives We have for F P L 2 Dx,y 2 F D x D y F D y D x F Dx n 1,...,x n F D x1 Dx n 1 2,...,x n 1 F f n px 1,..., x n q 1 n! ED x 1,...,x n F. measures the n-th order interactions between the points. (used for Poisson Voronoi approximation) For instance, f 2 px, yq 1 2 E pd xpd y F qq 1 E pf px Y tx, yu F px Y tyuq pf px Y txuq F px qqq 2 R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

27 Orstein-Uhlenbeck operator We introduce Under the expansion representation LF δdf LF ÿ ni n pf n q ně0 and its inverse on the space of centred functionals L 1 F ÿ ně0 1 n I npf n q R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

28 Stein-Malliavin bound (Peccati, Solé, Utzet, Taqqu 2010) It yields for F centred with variance 1 EFf pf q EpLL 1 F qf pf q EδDpL 1 F qf pf q ExDL 1 F, Df pf qy ExDL 1 F, f 1 pf qdf ` RpDF qy where RpDF q ď DF 2 because }f 2 } 8 ď 2. Then, with T xdl 1 F, DF y ˇ ˇEFf pf q f 1 pf qˇˇ ď ˇˇEpf ż 1 pf qqpt 1qˇˇ ` E ˇˇpD x F q 2 D L 1 x F ˇˇ µpdxq E ď c a ż varpt q ` E ˇˇpD x F q 2 D L 1 x F ˇˇ µpdxq E (ET varpf q 1) R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

29 Product formula and contractions In the formula above appear products of the form I m pf qi k pgq and we can prove that under technical assumptions ÿ I m pf qi k pgq κ r,l I m`k r`l pf l r gq 0ďlďrďm^k where f l r g is a contraction and has an explicit integral expression: ż f l r gpx m r, x 1 k r, y r lq f px m r, y r l, z l qgpx 1 k r y r l, z l qµ l pz l q Then if F U k phq ř 1ďmďk I mph m q, ˆ d W p F, Nq ď C k max }h m l r h n } 2 ` max k }h m} 2 1ďlďrďk,l k m 1 L loooooooooooooooooooooooooomoooooooooooooooooooooooooon 4 M R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

30 Application: 4th moment theorem (LR-Peccati 1, p.14) Assume F ř n k 0 I kpf k q with each f k ě 0, EF 0, EF 2 1. Then a cm} ď EF 4 3 ď CM Therefore if F 4 is uniformly integrable, it converges to N iff its 4 first moments do. example: Counting functionals R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

31 1 Stein s method 2 Stochastic Poisson integrals 3 Some geometric problems 4 Malliavin calculus 5 Example: Graph with geometric connections 6 Other limit theorems 7 Semi-group representations R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

32 Different regimes: H λ α λ H 1 H 1 : a 1, α λ 1, λ 25: y 1{y a 1{x a x CLT at speed λ 1{2 a 1, α λ λ 1{d, λ 50: CLT in logpλq 1 a 2, α λ λ 1{d, λ 50: no CLT. R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

33 Interaction volume We come back to the U-statistics examples. They are under the form ÿ ÿ U k phq hpx 1,..., x k q h 0 pα λ px 1,..., x k qq px 1,...,x k qpx k px 1,...,x k qpx k call v λ α d λ the interaction volume. Since h λpx k q h 0 pα λ x k q and h 0 is small far from the diagonal, v λ is the magnitude of the number of points having a significant influence on the contribution of a typical point. Subgraph counting: v λ α d λ Boolean model: v λ 1 (integrability condition on the random radii) Sylvester problem, flat processes: v λ λ R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

34 Assumptions X λ Poisson process with measure l1 λ 1{d W Scaled kernel: h λ px k q h 0 pα λ x k q h 0 is stationary and rapidly decreasing away from the diagonal: there is a probability density 0 ă κ ď C ă 8 such that for p 2, 4, ż h 0 p0, x k 1 q p κpx k 1 q 1 p dx k ă 8. h 0 has non-zero projections: for 1 ď m ď k ż x m ÞÑ h 0 px m, x k m qdx k m ı 0, R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

35 Main result (LR-Peccati) Then we have and calling varpu k ph λ qq 0 ă C 1 ď λv 2k 2 λ maxp1, v k`1 λ q ď C 2 ă 8 Ũ k ph λ q U kph λ q EU k ph λ q a varpuk ph λ qq we have where d is either dpũ k ph λ q, Nq ď C λ 1{2 3 maxp1, v pk 1q{2 λ q d W the Wasserstein distance d K the Kolmogorov distance (using Thaële & Eischelsbacher 2013) and N is a standard Gaussian variable R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

36 Regimes (c ą 0 constant) CLT Regimes: v λ " c: CLT with distance ď λ 1{2. Low order chaoses dominate. v λ c: CLT with distance ď λ 1{2. Chaoses co-dominate. λ 1{pk 1q! v λ! c: CLT with distance! 1{2. High order chaoses dominate. Applies to subgraph counting, flat processes, Sylvester problem, and segment intersections if Non-CLT Regimes: PplengthpK 1 q ě rq ď Cr 5 ε, r ě 1. v λ λ 1{pk 1q : Poisson limit. v λ! λ 1{pk 1q : Zero. R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

37 1 Stein s method 2 Stochastic Poisson integrals 3 Some geometric problems 4 Malliavin calculus 5 Example: Graph with geometric connections 6 Other limit theorems 7 Semi-group representations R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

38 Multidimensionnal CLTs (see p of Peccati and Zheng) Let F n pf 1,..., F q q where the F i s are Poisson functionals. Let N N p0, Cq where C is a m-dimensional covariance matrix. We have g f qÿ d H pf, Nq ďc e rc i,j ExDF i, DL 1 F j ys i,j 1 ż qÿ ` i 1 D x F i 2 qÿ i 1 where h P H if h P C 3 with }h 2 } 8, }h 3 } 8 ď 1. ˇ ˇD L 1 x F iˇˇ µpdxq R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

39 Poisson limits Let c ą 0 Theorem (Peccati 2012) If F P L 2 has integer values, then d TV pf, Ppcqq ď 1 e c E ˇˇc xdf, DL 1 F yˇˇ c ` 1 e c ˆż ˇ c 2 E ˇD x F pd x F 1qD L 1 x F ˇˇ Stein s equation for the Poisson law: If F has integer values, it is Ppcq iff it satisfies E Ecf px ` 1q Xf px q 0 If F does not have integer values, one introduces perturbations B such that F B P Z. R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

40 Poisson limit (II) If F n I q pf q,n q ` B n with q ě 2 and EF n Ñ c, varpf n q Ñ c, B n Ñ 0. If for 1 ď l ď r ď q, l q, and then d TV pf n, Ppcqq Ñ 0. }f n,q l r f n,q } L 2 Ñ 0 ż `f 2 n ` q! 2 fn 4 2q!fn 3 dµ q Ñ 0, R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

41 Portmanteau-inequality (Bourguin-Peccati 2012) X Ppcq N N p0, 1q F, G Poisson functionals, EF c, F P Z`, EG 0, varpgq 1 d H ppf, Gq,pX, Nqq ď C `E ˇˇc xdf, DL 1 F yˇˇ ` E ˇˇ1 xdg, DL 1 Gyˇˇ ż `E ˇˇDx F pd x F 1qD L 1 x F ˇˇ µpdxq ż `E D x G 2 ˇˇDx L 1 Gˇˇ µpdxq `ExˇˇDL 1 Gˇˇ, DF y where hpx, yq P H if h ď 1 and h is 1-Lipschitz in y. R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

42 1 Stein s method 2 Stochastic Poisson integrals 3 Some geometric problems 4 Malliavin calculus 5 Example: Graph with geometric connections 6 Other limit theorems 7 Semi-group representations R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

43 Semi-group representation and Glauber dynamics Consider the following dynamics on the Poisson measure X t, t ą 0: X 0 X Each point of the Poisson process disappears independently with rate 1. New points appear according to the rate µpdxqdt. Locally X t Ñ X 8 an independent Poisson measure as t Ñ 8. Given a functional f, we define the semigroup P t f pxq E 1 f px t X 0 xq for any locally finite point configuration x. R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

44 Mehler s formula (Privault,...) It turns out that for t ą 0, with T t P e t, ÿ P t I n pf n q ÿ e nt I n pf n q ně0 ně0 We have for F P L 2 centred (take F I q pf q) e qt 1 P t F F LF qi q pf q lim F lim ż tñ0 t tñ0 t L 1 F P t Fdt tą0 Remember that L 1 F n 1 F. And and therefore ż tą0 P t F e nt F P t Fdt L 1 F. R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

45 Last-Peccati-Schulte 2014 Second order Poincaré inequality d W pf, Nq ď γ 1 ` γ 2 ` γ 3 d K pf, Nq ď γ 1 ` γ 2 ` γ 3 ` γ 4 ` γ 5 ` γ 6 General inequalities and stabilization R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

46 Malliavin derivative over binomial input? Let X X 1,..., X n be IID variables, X 1 px 1 1,..., X 1 nq independent copies, and F F px 1,..., X n q a L 2 functional. Define where D i F px, X 1 q F px q F px i q X i px 1,..., X 1 i,..., X n q. Then F has the orthogonal (Hoeffding) decomposition nÿ ÿ F EF px q ` p 1q k E i1... ik F px q EF px q ` loooooooooooooooooooooomoooooooooooooooooooooon k 1 1ďi 1 ă ăi k ďn nÿ For ti 1,..., i k u tj 1,..., j m u, ÿ k 1 1ďi 1 ă ăi k ďn I k pf k q 2 f i1,...,i k px i1,..., X ik q Ep i1... ik F qp j1... jm F q 0. R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

47 Application: variance lower bound (symmetric case) varpf q ě ě n nÿ ErpErD i F px 1, X q X sq 2 s i 1 ż perf px q f px, X 2,..., X n qsq 2 µpdxq to be compared with Stein-Efron inequality varpf px qq ď n ż Erf px q f px, X 2,..., X n qs 2 2 The lower bound seems to be sharp when the problem is inhomogeneous (e.g. Voronoi set approximation) R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48

48 Survey book coming soon (?) R. Lachieze-Rey Stein-Malliavin method in the Poisson framework December 8, / 48