Stein’s method Motivation Discrete Malliavin calculus Convergence results Stein’s method and Malliavin calculus for independent random variables H´ el` ene Halconruy under the supervision of Laurent Decreusefond T´ el´ ecom ParisTech - Universit´ e de Paris Saclay bla bla 1/ 14
Stein’s method Motivation Discrete Malliavin calculus Convergence results Introduction Theorem (Central Limit Theorem) Let ( X n , n ≥ 1) be a sequence of i.i.d. twice integrable random variables defined on (Ω , A , P ) �� 1 � n √ n � � D − E [ X 1 ] − n →∞ N (0 , var[ X 1 ]) . − − − → X k n k =1 2/ 14
Stein’s method Motivation Discrete Malliavin calculus Convergence results Introduction Theorem (Central Limit Theorem) Let ( X n , n ≥ 1) be a sequence of i.i.d. twice integrable random variables defined on (Ω , A , P ) �� 1 � n √ n � � D − E [ X 1 ] − n →∞ N (0 , var[ X 1 ]) . − − − → X k n k =1 → Rate of convergence for the law of numbers. 2/ 14
Stein’s method Motivation Discrete Malliavin calculus Convergence results Introduction Theorem (Central Limit Theorem) Let ( X n , n ≥ 1) be a sequence of i.i.d. twice integrable random variables defined on (Ω , A , P ) �� 1 � n √ n � � D − E [ X 1 ] − n →∞ N (0 , var[ X 1 ]) . − − − → X k n k =1 � �� � F ( X 1 , ··· ,X n ) → Rate of convergence for the law of numbers. Question : how to estimate dist( F ∗ P n , P ) where P n := ⊗ n k =1 P X k and for P , Q measures on a Polish space F � � � � � � dist W ( P , Q ) = sup g d P − g d Q � � � � g ∈T and T = { g ∈ C 1 ( R , R ) : � g ′ � ∞ ≤ 1 } ? 2/ 14
Stein’s method Motivation Discrete Malliavin calculus Convergence results Introduction Theorem (Central Limit Theorem) Let ( X n , n ≥ 1) be a sequence of i.i.d. twice integrable random variables defined on (Ω , A , P ) �� 1 � n √ n � � D − E [ X 1 ] − n →∞ N (0 , var[ X 1 ]) . − − − → X k n k =1 � �� � F ( X 1 , ··· ,X n ) → Rate of convergence for the law of numbers. Question : how to estimate dist( F ∗ P n , P ) where P n := ⊗ n k =1 P X k and for P , Q measures on a Polish space F � � � � � � dist W ( P , Q ) = sup g d P − g d Q � � � � g ∈T and T = { g ∈ C 1 ( R , R ) : � g ′ � ∞ ≤ 1 } ? 2/ 14
Stein’s method Motivation Discrete Malliavin calculus Convergence results Stein’s method (1) • Caracterization of the target measure P . � For all g ∈ T , L g d Q = 0 ⇐ ⇒ Q = P . L is the Stein operator associated to P . 3/ 14
Stein’s method Motivation Discrete Malliavin calculus Convergence results Stein’s method (1) • Caracterization of the target measure P . � For all g ∈ T , L g d Q = 0 ⇐ ⇒ Q = P . L is the Stein operator associated to P . • Resolution of the Stein equation � L ϕ g = g − g d P , (1) for any g ∈ T , then g ∈ T ⇐ ⇒ ϕ g ∈ F . 3/ 14
Stein’s method Motivation Discrete Malliavin calculus Convergence results Stein’s method (1) • Caracterization of the target measure P . � For all g ∈ T , L g d Q = 0 ⇐ ⇒ Q = P . L is the Stein operator associated to P . • Resolution of the Stein equation � L ϕ g = g − g d P , (1) for any g ∈ T , then g ∈ T ⇐ ⇒ ϕ g ∈ F . • Equivalent problem � � � � � � g d Q − | E [ L ϕ ( X )] | where X ∼ Q . sup g d P � = sup � � � g ∈T ϕ ∈F 3/ 14
Stein’s method Motivation Discrete Malliavin calculus Convergence results Stein’s method (1) • Caracterization of the target measure P . � For all g ∈ T , L g d Q = 0 ⇐ ⇒ Q = P . L is the Stein operator associated to P . • Resolution of the Stein equation � L ϕ g = g − g d P , (1) for any g ∈ T , then g ∈ T ⇐ ⇒ ϕ g ∈ F . • Equivalent problem � � � � � � g d Q − | E [ L ϕ ( X )] | where X ∼ Q . sup g d P � = sup � � � g ∈T ϕ ∈F 3/ 14
Stein’s method Motivation Discrete Malliavin calculus Convergence results Stein’s method (2) • Principle : dist T ( P , Q ) = sup | E [ L ϕ ( X )] | ϕ ∈F 4/ 14
Stein’s method Motivation Discrete Malliavin calculus Convergence results Stein’s method (2) • Principle : dist T ( P , Q ) = sup | E [ L ϕ ( X )] | ϕ ∈F = sup | E [ L 1 ϕ ( X )] + E [ L 2 ϕ ( X )] | ϕ ∈F 4/ 14
Stein’s method Motivation Discrete Malliavin calculus Convergence results Stein’s method (2) • Principle : dist T ( P , Q ) = sup | E [ L ϕ ( X )] | ϕ ∈F = sup | E [ L 1 ϕ ( X )] + E [ L 2 ϕ ( X )] | ϕ ∈F • Idea : transform L 1 ϕ ( X ) into −L 2 ϕ ( X )+ remainder → rate of convergence. 3 methods : 1. Exchangeable pairs. 2. Size-biased. 3. Malliavin integration by parts. 4/ 14
Stein’s method Motivation Discrete Malliavin calculus Convergence results Stein’s method (2) • Principle : dist T ( P , Q ) = sup | E [ L ϕ ( X )] | ϕ ∈F = sup | E [ L 1 ϕ ( X )] + E [ L 2 ϕ ( X )] | ϕ ∈F • Idea : transform L 1 ϕ ( X ) into −L 2 ϕ ( X )+ remainder → rate of convergence. 3 methods : 1. Exchangeable pairs. 2. Size-biased. 3. Malliavin integration by parts. 4/ 14
Stein’s method Motivation Discrete Malliavin calculus Convergence results Finite dimension differential calculus and Malliavin calculus Analogous terminology Differential calculus on Malliavin calculus on Classical Euclidian spaces Wiener space Vectors Functions Gradient 5/ 14
Stein’s method Motivation Discrete Malliavin calculus Convergence results Finite dimension differential calculus and Malliavin calculus Analogous terminology Differential calculus on Malliavin calculus on Classical Euclidian spaces Wiener space Vectors Paths of Brownian motion Functions Gradient 5/ 14
Stein’s method Motivation Discrete Malliavin calculus Convergence results Finite dimension differential calculus and Malliavin calculus Analogous terminology Differential calculus on Malliavin calculus on Classical Euclidian spaces Wiener space Vectors Paths of Brownian motion Random variables Functions = Functionals of the paths Gradient 5/ 14
Stein’s method Motivation Discrete Malliavin calculus Convergence results Finite dimension differential calculus and Malliavin calculus Analogous terminology Differential calculus on Malliavin calculus on Classical Euclidian spaces Wiener space Vectors Paths of Brownian motion Random variables Functions = Functionals of the paths Gradient Malliavin derivative 5/ 14
Stein’s method Motivation Discrete Malliavin calculus Convergence results Stein-Malliavin criterion on the Gaussian space • H = L 2 ( T, B , µ ) real separable Hilbert space. • X = { X ( h ) , h ∈ H } centered I.G.P. and ˜ Q = X ∗ P . • S : space of cylindrical random variables of the form F = f ( X ( h 1 ) , . . . , X ( h n )) ; f ∈ C c ( R n , R ) , h i ∈ H. n ∂f � For F ∈ S , DF = ( X ( h 1 ) , . . . , X ( h n )) h i . ∂x i i =1 • δ adjoint of D and L = − δD ”Laplacian operator”. Theorem (Nourdin, Peccati) For any F ∈ D 1 , 2 with E [ F ] = 0 , � 2 �� 1 � �� 2 . � F ∗ ˜ � � � 1 − � DF , − DL − 1 F � H dist W Q , P ≤ E 6/ 14
Stein’s method Motivation Discrete Malliavin calculus Convergence results Sketch of the proof • L ϕ ( x ) = xϕ ( x ) − ϕ ′ ( x ) =: L 1 ϕ ( x ) + L 2 ϕ ( x ). • F = { ϕ ∈ C 2 ( R , R ) : � ϕ ′ � ∞ ≤ 1 , � ϕ ′′ � ∞ ≤ 2 } . L 1 ϕ ( F ) � �� � E F ϕ ( F ) ���� L ( L − 1 F ) 7/ 14
Stein’s method Motivation Discrete Malliavin calculus Convergence results Sketch of the proof • L ϕ ( x ) = xϕ ( x ) − ϕ ′ ( x ) =: L 1 ϕ ( x ) + L 2 ϕ ( x ). • F = { ϕ ∈ C 2 ( R , R ) : � ϕ ′ � ∞ ≤ 1 , � ϕ ′′ � ∞ ≤ 2 } . L 1 ϕ ( F ) � �� � � � − δ ( DL − 1 F ) ϕ ( F ) E F ϕ ( F ) = E ���� L ( L − 1 F ) 7/ 14
Stein’s method Motivation Discrete Malliavin calculus Convergence results Sketch of the proof • L ϕ ( x ) = xϕ ( x ) − ϕ ′ ( x ) =: L 1 ϕ ( x ) + L 2 ϕ ( x ). • F = { ϕ ∈ C 2 ( R , R ) : � ϕ ′ � ∞ ≤ 1 , � ϕ ′′ � ∞ ≤ 2 } . L 1 ϕ ( F ) � �� � � I.P.P. � � � − δ ( DL − 1 F ) ϕ ( F ) � Dϕ ( F ) , − DL − 1 F � H E F ϕ ( F ) = E = E ���� L ( L − 1 F ) 7/ 14
Stein’s method Motivation Discrete Malliavin calculus Convergence results Sketch of the proof • L ϕ ( x ) = xϕ ( x ) − ϕ ′ ( x ) =: L 1 ϕ ( x ) + L 2 ϕ ( x ). • F = { ϕ ∈ C 2 ( R , R ) : � ϕ ′ � ∞ ≤ 1 , � ϕ ′′ � ∞ ≤ 2 } . L 1 ϕ ( F ) � �� � � I.P.P. � � � − δ ( DL − 1 F ) ϕ ( F ) � Dϕ ( F ) , − DL − 1 F � H E F ϕ ( F ) = E = E ���� L ( L − 1 F ) � � ϕ ′ ( F ) � DF , − DL − 1 F � H = E (chain rule) 7/ 14
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