Kolmogorov Berry-Esseen bounds for binomial functionals

Transcription

1 Kolmogorov Berry-Esseen bounds for binomial functionals Raphaël Lachièze-Rey, Univ. South California, Univ. Paris 5 René Descartes, Joint work with Giovanni Peccati, University of Luxembourg, Singapore

2 Framework n 1 X = (X 1,..., X n ) IID variables on some space (E, E) (x 1,..., x n ) f (x 1,..., x n ) a real functional W = f (X ) has finite variance σ 2. N : Standard Gausian variable. We investigate here Note also d W (W, N) := sup Eh(W ) h(n) h 1 d K (W, N) := sup P(W t) P(N t) t R W = W EW Var(W )

3 1 A general bound 2 Hoeffding decomposition and variance lower bound 3 Geometric applications 4 Analogy with Poisson results

4 Stein s bound It is known that there is C > 0 such that for any suitable W d W (W, N) sup g: g C, g C, g C sup g: g C, g C,( ) t,t R EWg(W ) g (W ). Recent results by Schulte 2012 and Eischelsbacher & Thaele 2014 yield d K (W, N) EWg(W ) g (W ) where ( ) t replaces Taylor expansion at the 2d order g(x + u) g(x) ug (x) u2 2 ( x + C) + u ( 1 {x t<x+u} + 1 {x+u t<x} ) = u2 2 ( x + C) + u ( 1 {x t<x+u} 1 {x+u t<x} )

5 Some difference operators and a decomposition Introduce an additional vector X (d) = X independent of X and X A = (X A i ) 1 i n, A [n], and the difference operator X A i = { X i if i / A X i if i A, A f (X, X ) = f (X ) f (X A ) j f (X, X ) = {j} f (X, X ) Then, given two L 2 functionals ϕ, ψ, Chatterjee 2008 derived the decomposition cov(ϕ(x ), ψ(x )) = 1 1 ( 2 n E[ j ϕ(x, X A [n] A ) ) j ψ(x A, X )] (n A ) j / A }{{} κ n,a convex combination in the sense that A [n],j / A κ n,a = 1.

6 General bounds if EW = 0 This decomposition is the starting point for proving the bound d W (W, N) σ 2 2π n Var(E(T W )) + σ 3 E j f (X, X ) 3 16 j=1 }{{} B W d K (W, N) B W + σ 2 Var(E(T W )) + σ 4 4 E κ n,a f (X ) j f (X, X ) 2 j f (X A, X ) j,a,j / A where T = 1 2 T = 1 2 κ n,a j f (X, X ) j f (X A, X ), ET = σ 2 j,a:j / A j,a:j / A κ n,a j f (X, X ) j f (X A, X ), ET = 0,

7 In all the applications, T is bounded exactly like T In many applications, the other new term is bounded by σ 3 4 n j=1 E j f (X, X ) 6 (not optimal for Voronoi approximation) If not optimal and if f is symmetric, there is instead the bound n σ 4 4 sup E f (X ) 1 f (X A, X ) 3, A [n] As often, it requires lower bounds on σ.

8 1 A general bound 2 Hoeffding decomposition and variance lower bound 3 Geometric applications 4 Analogy with Poisson results

9 Hoeffding s decomposition It is known that any L 2 functional W = f (X ) admits a (unique) Hoeffding decomposition f (X ) = n f k (X ) k=0 where f k (X ) is a degenerate U-statistic of order k, that is f k (X ) = h i1,...,i k (X i1,..., X ik ) 1 i 1 <...i k n where the integral of any of the arguments of h i1,...,i k vanish, that is Eh(x 1,..., x i 1, X i, x i+1,..., x k ) = 0 for µ a.a. x j, j i. This implies the orthogonality Ef k (X )f j (X ) = 0, k j.

10 The orthogonality yields n Var(f (X )) = Var(f k (X )) k=1 The kernel h i1,...,i k operators: has a representation in terms of difference h i1,...,i k (x 1,..., x k ) = ( 1) k E i1 ( i2 (... ik f (X, (x 1,..., x k )))) The previous decomposition implies the variance lower bound Var(f (X )) Var(f 1 (X )) = n E [ (E j f (X ), X ) 2 X ] The form of the kernels recalls the orthogonal Wiener-Ito chaotic decomposition of Poisson functionals. j=1

11 1 A general bound 2 Hoeffding decomposition and variance lower bound 3 Geometric applications 4 Analogy with Poisson results

12 Geometric assumptions In many geometric applications, f can be seen as a functional taking in argument an unordered finite set of points. This has two formal consequences: f is symmetric in its arguments f can have any finite number of arguments. Introduce X î = (X j ; j i), and the derivatives D i f (X ) = f (X ) f (X î) D i,j f (X ) = D i (D j f (X )) = f (X ) f (X î) f (X ĵ) + f (X îĵ ) = D j,i f (X ). A bound that is useful on D i,j f (X ) is ( D i,j f (X ) D i f (X ) + D i f (X ĵ) ) ) 1 {Di,j f (X ) 0} where D i,j f (X ) 0 means somehow that i and j interact, i.e. the contribution of X i is affected by the presence of j, i.e. D i f (X ) D i f (X ĵ).

13 Example: The boolean model Let X 1,..., X n IID random compact sets in some space E R d, ( ) n f vol (X ) = vol E X i. i=1 Then D i f vol (X ) vol(x i ) D i,j f vol (X ) = 0 if X i X j =. f iso (X ) = #{j : X j X i E =, j i}, then D i f iso (X ) #{j : X j X i } D i,j f iso (X ) = 0 if no X k touches X i and X j.

14 Bound on Var(E[T W ]), Var(E[T W ]) Back to the general case Introduce X an independent copy of X (also independent of X ). Call recombination of {X, X, X } a vector Y = (Y 1,..., Y n ) where for each j, Y j {X j, X j, X j }. If f is symmetric, expanding Var(E(T W )) gives, up to some universal C > 0 Var(E[T ( ) W ]) C n sup j,k=1 (Y,Y,Z) E [1 {D1,j f (Y ) 0,D 1,k f (Y ) 0}D j f (Z) 4] where the supremum is over recombinations Y, Y, Z of {X, X, X } By symmetry, it suffices to treat the cases j = k = 1, j k = 1, j k 1

15 Results for the boolean model Assume E = n 1/d [0, 1] d and the X i are built in the following way: There are iid compact sets K 1,..., K n and IID points uniform Y 1,..., Y n in C n := n 1/d [0, 1] d such that X i = (Y i + K i ) E n. Then, if Ediam(K i ) 5d <, and if Ediam(K i ) 8d <, d K ( f vol (X ), N) Cn 1/2 d K ( f iso (X ), N) Cn 1/2. Goldstein and Penrose 2010 obtained similar results with the size-bias method.

16 Set estimation K is an unknown set in [0, 1]d X is a random sample of points We have the information {1x K, x X } How to get a good idea of K? Measure the quality of the approximation? R. Lachieze-Rey Kolmogorov Berry-Esseen bounds for binomial functionals

17 Voronoi approximation Reconstruct K with K X = {x Rd : x is closer from a point of X inside K than outside K }. R. Lachieze-Rey Kolmogorov Berry-Esseen bounds for binomial functionals

18 Regularity hypotheses Many applications (cardiology, oncology, chemistry) are concerned with the detection of irregular boundaries. We developed a general method allowing the set to be irregular in some sense. Assume that The tubular neighborhood of K s boundary satisfies for some α > 0, 0 < C C + < C r α Vol( K B(0, r)) C + r α Around most points of the boundary, is not too small. Vol(K c B(x, r)) r d These assumptions are satisfied by some fractal sets, such as for instance the Von Koch flake and anti flake, and also by some smooth sets which boundaries are piecewise C 1.

19 Derivatives and second-order derivatives When one removes the point X 1 from X, the approximation Vol(K X ) can only vary by the volume of the Voronoi cell Vor(X 1 ; X ) of X 1 of X, and it does so only if X 1 is close to the boundary. D 1 f (X ) Vol(Vor(X 1 ; X ))1 {X is at Voronoi-distance less than 2 from K} Also, two points X i, X j interact if they are at Voronoi-distance less than 2 and one of them is at distance less than 2 from K, which gives a sharp bound on 1 {Di,j f (X ) 0}.

20 Second order theory Given K satisfying the previous regularity hypotheses, the variance and Berry-Esseen bounds presented above apply and we have for some C, C, C > 0, and Cn 1 α/d Var(Vol(K X )) C n 1 α/d d K (Ṽol(K X ), N) C n 1/2+α/2d log(n) 3+α/d+ε. Consistent with Yukich results with Poisson input and sets with a d 1-dimensional boundary.

21 1 A general bound 2 Hoeffding decomposition and variance lower bound 3 Geometric applications 4 Analogy with Poisson results

22 Second order Poincaré inequality The bound involving second order derivatives bears some similarity with the second order Poincare inequality derived by Last, Peccati, and Schulte (2014), where for some Poisson process X and L 2 functional f (X ), d K ( f (X ), N) 6 i=1 γ i where the γ i are estimates depending on the first and second add-one cost derivatives, for instance [ [ γ 1 = 4 E (D x1f ) 2 (D F )2] 1/2 [ x2 E(D 2 x 1,x 3 F ) 2 (Dx 2 2,x 3 F ) 2] ] 1/2 dx1 dx 2 dx 3 The first Second-order Poincaré inequality was derived by Chatterjee 2009 in the Gaussian framework, and generalized by Nourdin and Peccati (2009)

23 Variance bounds Poisson input X with intensity µ: Var(f (X )) is between (E [f (X x) f (X )]) 2 µ(dx) and E(f (X x) f (X )) 2 µ(dx) }{{}}{{} First term of the orthogonal chaos decomposition Poincaré inequality Binomial input X : Define X îx the vector X where the i-th component has been replaced by x. Then Var(f (X )) is between n i=1 ( E [ f (X îx) f (X )]) 2 µ(dx) } {{ } 1t term of Hoeffding decomposition and 1 2 n i=1 ( E f (X îx) 2 f (X )) µ(dx) } {{ } Stein-Efron inequality