Plan Stein’s method for normal approximation Applications Other target distributions : invariant measures of diffusions Examples Fouth Moment Theorem The case when the diffusion coefficient is a polynomial of second degree Stein’s method and Malliavin calculus Ciprian A. Tudor Universit´ e de Lille 1 International Colloquim on Stein’s method, Concentration Inequalities and Malliavin calculus Missillac, France June 2014 Ciprian A. Tudor Stein’s method and Malliavin calculus
Plan Stein’s method for normal approximation Applications Other target distributions : invariant measures of diffusions Examples Fouth Moment Theorem The case when the diffusion coefficient is a polynomial of second degree Stein’s method for normal approximation 1 2 Applications 3 Other target distributions : invariant measures of diffusions 4 Examples 5 Fouth Moment Theorem 6 The case when the diffusion coefficient is a polynomial of second degree Ciprian A. Tudor Stein’s method and Malliavin calculus
Plan Stein’s method for normal approximation Applications Other target distributions : invariant measures of diffusions Examples Fouth Moment Theorem The case when the diffusion coefficient is a polynomial of second degree The purpose of the so-called Stein method is to measure the distance between two probability distributions. This distance, denoted by d , can be defined in several ways : the Kolmogorov distance the Wasserstein distance the total variation distance the Fortet-Mourier distance. Ciprian A. Tudor Stein’s method and Malliavin calculus
Plan Stein’s method for normal approximation Applications Other target distributions : invariant measures of diffusions Examples Fouth Moment Theorem The case when the diffusion coefficient is a polynomial of second degree Concretely, let X , Y be two random variables. The distance between the law of X and the law of Y is usually defined by ( L ( F ) denotes the law of F ) d ( L ( X ) , L ( Y )) = sup | Eh ( X ) − Eh ( Y ) | h ∈H where H is a suitable class of functions. Ciprian A. Tudor Stein’s method and Malliavin calculus
Plan Stein’s method for normal approximation Applications Other target distributions : invariant measures of diffusions Examples Fouth Moment Theorem The case when the diffusion coefficient is a polynomial of second degree For example, if H is the set of indicator function 1 ( −∞ , z ] , z ∈ R we obtain the Kolmogorov distance d K ( L ( X ) , L ( Y )) = sup | P ( X ≤ z ) − P ( Y ≤ z ) | . z ∈ R If H is the set of 1 B with B a Borel set, one has the total variation distance d TV ( L ( X ) , L ( Y )) = sup | P ( X ∈ B ) − P ( Y ∈ B ) | . B ∈B ( R ) Ciprian A. Tudor Stein’s method and Malliavin calculus
Plan Stein’s method for normal approximation Applications Other target distributions : invariant measures of diffusions Examples Fouth Moment Theorem The case when the diffusion coefficient is a polynomial of second degree If H = { h ; � h � L ≤ 1 } ( � · � L is the Lipschitz norm) one has the Wasserstein distance. Other examples of distances between the distributions of random variables exist, e.g. the Fortet-Mourier distance. Ciprian A. Tudor Stein’s method and Malliavin calculus
Plan Stein’s method for normal approximation Applications Other target distributions : invariant measures of diffusions Examples Fouth Moment Theorem The case when the diffusion coefficient is a polynomial of second degree An important particular case : compute the distance between the law of an arbitrary r.v. F and the standard normal law Useful for many applications. For instance, in statistics, if an estimator is asymptotically normal, one needs to know how fast it converges to the normal distribution Ciprian A. Tudor Stein’s method and Malliavin calculus
Plan Stein’s method for normal approximation Applications Other target distributions : invariant measures of diffusions Examples Fouth Moment Theorem The case when the diffusion coefficient is a polynomial of second degree Let Z a r.v. with law N(0,1). How to estimate d ( F , Z ) = sup h ∈H | Eh ( F ) − Eh ( Z ) | In particular, how to compute the Kolmogorov distance sup | P ( X ≤ z ) − P ( Y ≤ z ) | z ∈ R Ciprian A. Tudor Stein’s method and Malliavin calculus
Plan Stein’s method for normal approximation Applications Other target distributions : invariant measures of diffusions Examples Fouth Moment Theorem The case when the diffusion coefficient is a polynomial of second degree The starting point to compute the distance between the law of F (an arbitrary r.v. ) and the law of Z is the Stein equation h ( x ) − Eh ( Z ) = f ′ ( x ) − xf ( x ) . h is given one needs to find the function f which is the solution of the Stein equation. in the case of the Kolmogorov distance, h ( x ) = 1 ( −∞ , z ] ( x ). Need to find f = f z that satisfies the Stein’s equation for every x . Ciprian A. Tudor Stein’s method and Malliavin calculus
Plan Stein’s method for normal approximation Applications Other target distributions : invariant measures of diffusions Examples Fouth Moment Theorem The case when the diffusion coefficient is a polynomial of second degree (F is arbitrary, Z ∼ N (0 , 1)) In the case of the Kolmogorov distance (in the Stein equation, put x = F and then take the expectation) � Ef ′ � � sup | P ( F < z ) − P ( Z < z ) | = sup z ∈ R z ( F ) − Ff z ( F ) � z ∈ R where f z is the solution of the Stein equation 1 ( −∞ , z ) ( x ) − P ( Z < z ) = f ′ ( x ) − f ( x ) , x ∈ R Key fact : the solution of the Stein equation is ”nice” (for example its derivative is bounded) Ciprian A. Tudor Stein’s method and Malliavin calculus
Plan Stein’s method for normal approximation Applications Other target distributions : invariant measures of diffusions Examples Fouth Moment Theorem The case when the diffusion coefficient is a polynomial of second degree Recall : we need to compute Ef ′ ( F ) − EFf ( F ) Idea : use some integration by parts to write EFf ( F ) = Ef ′ ( F ) G F Then Ef ′ ( F ) − EFf ( F ) = Ef ′ ( F )(1 − G F ) and use the fact that f ′ is ”nice” Ciprian A. Tudor Stein’s method and Malliavin calculus
Plan Stein’s method for normal approximation Applications Other target distributions : invariant measures of diffusions Examples Fouth Moment Theorem The case when the diffusion coefficient is a polynomial of second degree How to express G F ? The Malliavin calculus comes into the play ! The fundamental formula : if F is centered, then F = δ D ( − L ) − 1 F where D is the Malliavin derivative, L the Ornstein-Uhlenbeck operator, δ the divergence (Skorohod) integral Ciprian A. Tudor Stein’s method and Malliavin calculus
Plan Stein’s method for normal approximation Applications Other target distributions : invariant measures of diffusions Examples Fouth Moment Theorem The case when the diffusion coefficient is a polynomial of second degree How are these operators defined ? Let ’s undesrtand how they act on multiple stochastic integrals Let ( W t ) t ∈ [0 , 1] a standard Wiener process and I n the multiple integral of order n w.r.t. W . I n is an isometry from L 2 [0 , 1] n onto L 2 (Ω) EI n ( f ) 2 = n ! � ˜ f � 2 L 2 [0 , 1] n where ˜ f is the symmetrization of f I n ( f ) is also an iterated Itˆ o integral If f is symmetric, � 1 � t 2 I n = n ! dW t n . . . ..... dW t 1 f ( t 1 , . . . , t n ) 0 0 Ciprian A. Tudor Stein’s method and Malliavin calculus
Plan Stein’s method for normal approximation Applications Other target distributions : invariant measures of diffusions Examples Fouth Moment Theorem The case when the diffusion coefficient is a polynomial of second degree Wiener chaos decomposition : any random variable F ∈ L 2 (Ω , F , P ) ( F is the sigma-algebra generated by W ) can be written as � F = I n ( f n ) n ≥ 0 S [0 , 1] n (uniquely determined by F ) with f n ∈ L 2 S [0 , 1] n is called the the subset of L 2 (Ω) generated by I n ( f ) , f ∈∈ L 2 Wiener chaos of order n In Malliavin calculus, the multiple integrals are very useful (fit well with the theory) Ciprian A. Tudor Stein’s method and Malliavin calculus
Plan Stein’s method for normal approximation Applications Other target distributions : invariant measures of diffusions Examples Fouth Moment Theorem The case when the diffusion coefficient is a polynomial of second degree The Malliavin operators on Wiener chaos : D s I n ( f ) = nI n − 1 f ( · , s ) ( − L ) − 1 I n ( f ) = 1 nI n ( f ) δ I n f ( · , t ) = I n +1 (˜ f ) . Easy to see that F = δ D ( − L ) − 1 F if F = I n ( f ) Ciprian A. Tudor Stein’s method and Malliavin calculus
Plan Stein’s method for normal approximation Applications Other target distributions : invariant measures of diffusions Examples Fouth Moment Theorem The case when the diffusion coefficient is a polynomial of second degree Since EFf ( F ) = E δ D ( − L ) − 1 Ff ( F ) = Ef ′ ( F ) � D ( − L ) − 1 F , DF � so Ef ′ ( F ) − EFf ( F ) = E ( f ′ ( F )(1 − � D ( − L ) − 1 F , DF � ) Use Chauchy-Schwarz and remember that the derivative of the solution to the Stein equation is bounded. Ciprian A. Tudor Stein’s method and Malliavin calculus
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