Functional Steins Institut method Mines-Telecom L. Decreusefond - PowerPoint PPT Presentation

Functional Stein’s Institut method Mines-Telecom L. Decreusefond Borchard symposium

Roadmap Probability Optimal Types Transportation metrics Rubinstein Wasserstein Entropy Prohorov 2/39 Institut Mines-Telecom Functional Stein’s method

Roadmap Probability Optimal Types Transportation metrics Rubinstein Wasserstein Entropy Prohorov Applications Ergodicity Functional ineq. Convergence rate 2/39 Institut Mines-Telecom Functional Stein’s method

Roadmap Girsanov ν ≪ µ Probability Computations metrics Stein Applications ν = T ∗ m Ergodicity Optimal Types Functional Transportation ineq. Rubinstein Convergence rate Wasserstein Entropy Prohorov 2/39 Institut Mines-Telecom Functional Stein’s method

Roadmap Girsanov ν ≪ µ Probability Computations metrics Stein Applications ν = T ∗ m Ergodicity Optimal Types Functional Transportation ineq. Rubinstein Convergence rate Wasserstein Entropy Prohorov 2/39 Institut Mines-Telecom Functional Stein’s method

Rubinstein distance Definition ◮ ( F , d ) a Polish space 3/39 Institut Mines-Telecom Functional Stein’s method

Rubinstein distance Definition ◮ ( F , d ) a Polish space ◮ c a distance on F 3/39 Institut Mines-Telecom Functional Stein’s method

Rubinstein distance Definition ◮ ( F , d ) a Polish space ◮ c a distance on F ◮ F ∈ Lip c iff | F ( x ) − F ( y ) | ≤ c ( x , y ) 3/39 Institut Mines-Telecom Functional Stein’s method

Rubinstein distance Definition ◮ ( F , d ) a Polish space ◮ c a distance on F ◮ F ∈ Lip c iff | F ( x ) − F ( y ) | ≤ c ( x , y ) ◮ µ and ν ′ 2 proba. measures on F � � d R ( µ, ν ′ ) = sup F d ν ′ F d µ − F ∈ Lip c 3/39 Institut Mines-Telecom Functional Stein’s method

Some examples ◮ F = R n ◮ d = c =Euclidean distance ◮ Convergence in Rubinstein is equivalent to convergence in distribution (Dudley) 4/39 Institut Mines-Telecom Functional Stein’s method

Some examples ◮ F = R n ◮ d = c =Euclidean distance ◮ Convergence in Rubinstein is equivalent to convergence in distribution (Dudley) Wiener space ◮ F = space of continuous functions on [0 , 1] ◮ d =uniform distance ◮ c =distance in the Cameron-Martin space �� 1 � 1 / 2 | f ′ ( s ) − g ′ ( s ) | 2 d s c ( f , g ) = 0 4/39 Institut Mines-Telecom Functional Stein’s method

Configuration space Definition A configuration is a locally finite set of particles on a Polish space Y � � f d ω = f ( x ) x ∈ ω 5/39 Institut Mines-Telecom Functional Stein’s method

Configuration space Definition A configuration is a locally finite set of particles on a Polish space Y � � f d ω = f ( x ) x ∈ ω Vague topology � � vaguely n →∞ ω n − − − − → ω ⇐ ⇒ f d ω n − − − → f d ω for all f continuous with compact support from Y to R 5/39 Institut Mines-Telecom Functional Stein’s method

Configuration space Definition A configuration is a locally finite set of particles on a Polish space Y � � f d ω = f ( x ) x ∈ ω Vague topology � � vaguely n →∞ ω n − − − − → ω ⇐ ⇒ f d ω n − − − → f d ω for all f continuous with compact support from Y to R d is the associated distance 5/39 Institut Mines-Telecom Functional Stein’s method

Lipschitz functionals Distance between configurations c ( ω, η ) = dist TV ( ω, η ) = number of different points 6/39 Institut Mines-Telecom Functional Stein’s method

Lipschitz functionals Distance between configurations c ( ω, η ) = dist TV ( ω, η ) = number of different points Definition F : N Y → R is TV–Lip 1 if | F ( ω ) − F ( η ) | ≤ dist TV ( ω, η ) 6/39 Institut Mines-Telecom Functional Stein’s method

Lipschitz functionals Distance between configurations c ( ω, η ) = dist TV ( ω, η ) = number of different points Definition F : N Y → R is TV–Lip 1 if | F ( ω ) − F ( η ) | ≤ dist TV ( ω, η ) Definition (Rubinstein distance) � � E P F − E Q F d R ( P , Q ) := sup , F ∈ TV–Lip 1 6/39 Institut Mines-Telecom Functional Stein’s method

Convergence in configuration space Theorem (D- Schulte-Th¨ ale) d R ( P n , Q ) n →∞ distr. − − − → 0 = ⇒ P n − − − → Q Convergence in N Y 7/39 Institut Mines-Telecom Functional Stein’s method

Generic scheme Initial space Target space ( E , ν ) ( F , µ ) 8/39 Institut Mines-Telecom Functional Stein’s method

Generic scheme Initial space Target space ( E , ν ) ( F , µ ) T ( F , T ∗ ν ) 8/39 Institut Mines-Telecom Functional Stein’s method

Generic scheme Initial space Target space ( E , ν ) ( F , µ ) T dist.( T ∗ ν, µ ) ? ( F , T ∗ ν ) 8/39 Institut Mines-Telecom Functional Stein’s method

Motivation : Poisson polytopes 1 5 3 1 6 5 234 4 0 2 6 η ξ ( η ) 9/39 Institut Mines-Telecom Functional Stein’s method

Motivation : Poisson polytopes 1 3 5 1 6 5 234 4 0 2 6 η ξ ( η ) Question What happens when the number of points goes to infinity ? 9/39 Institut Mines-Telecom Functional Stein’s method

Poisson point process Hypothesis Points are distributed to a Poisson process of control measure t K Definition ◮ The number of points is a Poisson rv ( t K ( S d − 1 )) ◮ Given the number of points, they are independently drawn with distribution K 10/39 Institut Mines-Telecom Functional Stein’s method

Framework Initial space Target space ( N X , π t K ) ( N Y , π M ) 11/39 Institut Mines-Telecom Functional Stein’s method

Framework Initial space Target space ( N X , π t K ) ( N Y , π M ) T t ( N Y , T ∗ t π t K ) 11/39 Institut Mines-Telecom Functional Stein’s method

Framework Initial space Target space ( N X , π t K ) ( N Y , π M ) T t T t ( η ) = t γ � � = � x − y � ( N Y , T ∗ t π t K ) x , y ∈ η (2) 2 11/39 Institut Mines-Telecom Functional Stein’s method

Rescaling : γ = 2 / ( d − 1) Campbell-Mecke formula � � f ( x ) = t k E f ( x 1 , · · · , x k ) K (d x 1 , · · · , d x k ) x 1 , ··· , x k ∈ ω ( k ) � = 12/39 Institut Mines-Telecom Functional Stein’s method

Rescaling : γ = 2 / ( d − 1) Campbell-Mecke formula � � f ( x ) = t k E f ( x 1 , · · · , x k ) K (d x 1 , · · · , d x k ) x 1 , ··· , x k ∈ ω ( k ) � = Mean number of points (after rescaling) 1 � t γ x − t γ y �≤ β = t 2 1 �� � 2 E S d − 1 ⊗ S d − 1 1 � x − y �≤ t − γ β d x d y 2 x � = y ∈ ω 12/39 Institut Mines-Telecom Functional Stein’s method

Rescaling : γ = 2 / ( d − 1) Campbell-Mecke formula � � f ( x ) = t k E f ( x 1 , · · · , x k ) K (d x 1 , · · · , d x k ) x 1 , ··· , x k ∈ ω ( k ) � = Mean number of points (after rescaling) 1 � t γ x − t γ y �≤ β = t 2 1 �� � 2 E S d − 1 ⊗ S d − 1 1 � x − y �≤ t − γ β d x d y 2 x � = y ∈ ω Geometry β t − γ ( y )) = κ d − 1 ( β t − γ ) d − 1 +( d − 1) κ d − 1 V d − 1 ( S d − 1 ∩ B d ( β t − γ ) d + O ( t − γ ( 2 12/39 Institut Mines-Telecom Functional Stein’s method

Rescaling : γ = 2 / ( d − 1) Campbell-Mecke formula � � f ( x ) = t k E f ( x 1 , · · · , x k ) K (d x 1 , · · · , d x k ) x 1 , ··· , x k ∈ ω ( k ) � = Mean number of points (after rescaling) 1 � t γ x − t γ y �≤ β = t 2 1 �� � 2 E S d − 1 ⊗ S d − 1 1 � x − y �≤ t − γ β d x d y 2 x � = y ∈ ω Geometry β t − γ ( y )) = κ d − 1 ( β t − γ ) d − 1 +( d − 1) κ d − 1 V d − 1 ( S d − 1 ∩ B d ( β t − γ ) d + O ( t − γ ( 2 12/39 Institut Mines-Telecom Functional Stein’s method

Schulte-Th¨ ale (2012) based on Peccati (2011) � � P ( t 2 / ( d − 1) T m ( ξ ) > x ) − e − β x ( d − 1) m − 1 � ( β x d − 1 ) i � � � � � i ! � � i =0 � ≤ C x t − min(1 / 2 , 2 / ( d − 1)) 13/39 Institut Mines-Telecom Functional Stein’s method

Schulte-Th¨ ale (2012) based on Peccati (2011) � � P ( t 2 / ( d − 1) T m ( ξ ) > x ) − e − β x ( d − 1) m − 1 � ( β x d − 1 ) i � � � � � i ! � � i =0 � ≤ C x t − min(1 / 2 , 2 / ( d − 1)) What about speed of convergence as a process ? 13/39 Institut Mines-Telecom Functional Stein’s method

Stein method in one picture m = T ∗ ν P ∗ t m µ µ P ∗ t µ = µ 14/39 Institut Mines-Telecom Functional Stein’s method

Stein method The main tool Construct a Markov process ( X ( s ) , s ≥ 0) ◮ with values in F ◮ ergodic with µ as invariant distribution X ( s ) distr . − − − → µ for all initial condition X (0) ◮ for which µ is a stationary distribution X (0) distr. ⇒ X ( s ) distr. = µ = = µ, ∀ s > 0 ◮ Equivalently � LF d m = 0 , ∀ F iff m = µ 15/39 Institut Mines-Telecom Functional Stein’s method

Dirichlet structure LF = dPt F � � dt � t =0 LF / G tq E ( F , G )= Pt = etL E ( F , H )= d dt � Pt F , G � L 2( µ ) � G , H � L 2( m µ ) , ∀ H Pt F → � F d µ at t = 0 E ( F , G ) = � LF , G � L 2( µ ) µ, L � LF d µ = 0 16/39 Institut Mines-Telecom Functional Stein’s method