An Indexed Central Limit Theorem Bob Lowen (with B. Berckmoes, J. - PowerPoint PPT Presentation

An Indexed Central Limit Theorem An Indexed Central Limit Theorem Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp Coimbra July 2012 Workshop on Category Theory in honour of George Janelidze on the occasion of his 60th birthday Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

An Indexed Central Limit Theorem Standard triangular arrays Standard triangular array (STA): a triangular array of real square integrable random variables ξ 1 , 1 ξ 2 , 1 ξ 2 , 2 ξ 3 , 1 ξ 3 , 2 ξ 3 , 3 . . . satisfying the following properties. ( a ) ∀ n : ξ n , 1 , . . . , ξ n , n are independent ( b ) ∀ n , k : E [ ξ n , k ] = 0 n � σ 2 n , k = 1 , where σ 2 ξ 2 � � ( c ) ∀ n : n , k = E n , k k =1 n σ 2 ( d ) max n , k → 0 k =1 Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

An Indexed Central Limit Theorem The Lindeberg-Feller CLT Theorem Given an STA ( ξ n , k ) n , k and a normally distributed random variable ξ : if n � ξ 2 � � ∀ ǫ > 0 : n , k ; | ξ n , k | ≥ ǫ → 0 E k =1 then n w � ξ n , k → ξ. k =1 Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

An Indexed Central Limit Theorem Lindeberg index n � ξ 2 � � Lin ( { ξ n , k } ) = sup lim sup n , k ; | ξ n , k | ≥ ǫ E ǫ> 0 n →∞ k =1 Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

An Indexed Central Limit Theorem Lindeberg index n � ξ 2 � � Lin ( { ξ n , k } ) = sup lim sup n , k ; | ξ n , k | ≥ ǫ E ǫ> 0 n →∞ k =1 α Fix 0 < α < 1, let β = 1 − α and put n n k − 1 = n + β s 2 � � 1 − k − 1 � � n = (1 + β ) n − β k =1 k =1 P [ η α, n , k = − 1 / s n ] = P [ η α, n , k = 1 / s n ] = 1 1 − β k − 1 � � 2 √ √ = 1 � � � � 2 β k − 1 η α, n , k = − k / s n = P η α, n , k = k / s n P Lin ( { η α, n , k } ) = α Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

An Indexed Central Limit Theorem Steps in the proof � P [ η ≤ x ] − P [ η ′ ≤ x ] η, η ′ � � � � K = sup � x ∈ R Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

An Indexed Central Limit Theorem Steps in the proof � P [ η ≤ x ] − P [ η ′ ≤ x ] η, η ′ � � � � K = sup � x ∈ R Stein’s method (Stein 1986), Chen, Goldstein, Shao (Normal approximation by Stein’s method, Springer 2011) Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

An Indexed Central Limit Theorem Steps in the proof � P [ η ≤ x ] − P [ η ′ ≤ x ] η, η ′ � � � � K = sup � x ∈ R Stein’s method (Stein 1986), Chen, Goldstein, Shao (Normal approximation by Stein’s method, Springer 2011) H : all strictly decreasing functions h : R → R , bounded first and second derivatives and a bounded and piecewise continuous third derivative, x →−∞ h ( x ) = 1 and lim lim x →∞ h ( x ) = 0. Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

An Indexed Central Limit Theorem Steps in the proof � P [ η ≤ x ] − P [ η ′ ≤ x ] η, η ′ � � � � K = sup � x ∈ R Stein’s method (Stein 1986), Chen, Goldstein, Shao (Normal approximation by Stein’s method, Springer 2011) H : all strictly decreasing functions h : R → R , bounded first and second derivatives and a bounded and piecewise continuous third derivative, x →−∞ h ( x ) = 1 and lim lim x →∞ h ( x ) = 0. Step 1 If η is continuously distributed, then the formula lim sup n →∞ K ( η, η n ) = sup lim sup n →∞ | E [ h ( η ) − h ( η n )] | h ∈H is valid for any sequence ( η n ) n Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

An Indexed Central Limit Theorem Step 2 Let h : R → R be measurable and bounded. Put � x f h ( x ) = e x 2 / 2 ( h ( t ) − E [ h ( ξ )]) e − t 2 / 2 dt −∞ Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

An Indexed Central Limit Theorem Step 2 Let h : R → R be measurable and bounded. Put � x f h ( x ) = e x 2 / 2 ( h ( t ) − E [ h ( ξ )]) e − t 2 / 2 dt −∞ (Basic points of Stein’s method) (1) For any x ∈ R E [ h ( ξ )] − h ( x ) = xf h ( x ) − f ′ h ( x ) . (2) Moreover, � f ′′ � h ′ � � � � ∞ ≤ 2 ∞ , � � h (3) If h z = 1 ] −∞ , z ] for z ∈ R , then for all x , y ∈ R � ≤ 1 . � � f ′ h z ( x ) − f ′ � h z ( y ) Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

An Indexed Central Limit Theorem Step 3 Let h ∈ H and put       �  − f h �  − ξ n , k f ′ � δ n , k = f h ξ n , i + ξ n , k ξ n , i ξ n , i h  i � = k i � = k i � = k       ǫ n , k = f ′ �  − f ′ �  − ξ n , k f ′′ � ξ n , i + ξ n , k ξ n , i ξ n , i h h h  i � = k i � = k i � = k Then �� n � n � n � � �� � � � − f ′ ξ n , k f h ξ n , k ξ n , k E h k =1 k =1 k =1 Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

An Indexed Central Limit Theorem Step 3 Let h ∈ H and put       �  − f h �  − ξ n , k f ′ � δ n , k = f h ξ n , i + ξ n , k ξ n , i ξ n , i h  i � = k i � = k i � = k       ǫ n , k = f ′ �  − f ′ �  − ξ n , k f ′′ � ξ n , i + ξ n , k ξ n , i ξ n , i h h h  i � = k i � = k i � = k Then �� n � n � n � � �� � � � − f ′ ξ n , k f h ξ n , k ξ n , k E h k =1 k =1 k =1 n n � � σ 2 = E [ ξ n , k δ n , k ] − n , k E [ ǫ n , k ] k =1 k =1 Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

An Indexed Central Limit Theorem Step 4 Let f : R → R have a bounded derivative and a bounded and piecewise continuous second derivative. Then for any a , x ∈ R � � f ( a + x ) − f ( a ) − f ′ ( a ) x � � �� � � | x | , 1 � f ′ ( x 1 ) − f ′ ( x 2 ) � f ′′ � ∞ x 2 � � � ≤ min sup � � 2 x 1 , x 2 ∈ R Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

An Indexed Central Limit Theorem Step 4 Let f : R → R have a bounded derivative and a bounded and piecewise continuous second derivative. Then for any a , x ∈ R � � f ( a + x ) − f ( a ) − f ′ ( a ) x � � �� � � | x | , 1 � f ′ ( x 1 ) − f ′ ( x 2 ) � f ′′ � ∞ x 2 � � � ≤ min sup � � 2 x 1 , x 2 ∈ R Step 5 Let h ∈ H . Then for all x , y ∈ R � ≤ 1 � � f ′ h ( x ) − f ′ � h ( y ) Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

An Indexed Central Limit Theorem Step 6 n � | E [ h ( ξ ) − h ( ξ n , k )] | ≤ · · · k =1 ≤ · · · n ≤ 1 | ξ n , k | 3 ; | ξ n , k | < ǫ � � � f ′′ � � � E � h ∞ 2 k =1 � � n � | ξ n , k | 2 ; | ξ n , k | ≥ ǫ � � � � f ′ h ( x 1 ) − f ′ � + sup h ( x 2 ) E � x 1 , x 2 ∈ R k =1 � � n � f ′′ h ( x 1 ) − f ′′ � σ 2 � � + sup h ( x 2 ) n , k E [ | ξ n , k | ] � x 1 , x 2 ∈ R k =1 ≤ · · · Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

An Indexed Central Limit Theorem An inequality Theorem Given an STA ( ξ n , k ) n , k and a normally distributed ξ the inequality � n � � lim sup n →∞ K ξ, ξ n , k ≤ Lin ( { ξ n , k } ) k =1 is valid. Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

An Indexed Central Limit Theorem An inequality Theorem Given an STA ( ξ n , k ) n , k and a normally distributed ξ the inequality � n � � lim sup n →∞ K ξ, ξ n , k ≤ Lin ( { ξ n , k } ) k =1 is valid. ? Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

An Indexed Central Limit Theorem Enter approach theory F : probability distributions F c : continuous probability distributions ∗ : convolution Bergstr¨ om’s direct convolution method η n → η (weak) ⇔ ∀ ζ ∈ F c : η n ∗ ζ → η ∗ ζ (uniformly) Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem

An Indexed Central Limit Theorem Enter approach theory F : probability distributions F c : continuous probability distributions ∗ : convolution Bergstr¨ om’s direct convolution method η n → η (weak) ⇔ ∀ ζ ∈ F c : η n ∗ ζ → η ∗ ζ (uniformly) In Top (( F , T w ) → ( F , T K ) : η �→ η ∗ ζ ) ζ ∈F c Bob Lowen (with B. Berckmoes, J. Van Casteren) University of Antwerp An Indexed Central Limit Theorem