Central Limit Theorem for Analitic Functions of two-sided moving - PowerPoint PPT Presentation

Central Limit Theorem for Analitic Functions of two-sided moving averages. Limit theorem for canonical U -statistics Sidorov D. I., Novosibirsk State University 2nd Northern Triangular Seminar, 2010

Let { ξ j } j ∈ Z be a stationary sequence, and { a j ; j ∈ Z } be real num- bers. Moving-average sequence ( linear process ) is defined by � X k := (1) a k − j ξ j . j ∈ Z If � E | ξ 0 | < ∞ and | a j | < ∞ or (2) j ∈ Z E ξ 2 a 2 � { ξ j } j ∈ Z are i.i.d., E ξ 0 = 0 , 0 < ∞ , and j < ∞ , (3) j ∈ Z then (1) is well defined.

Outline 1. Mixing conditions 2. CLT for the sequence { g ( X k ) } k ≥ 1 , where g ( x ) is a non-linear function. 3. Limit theorem for canonical U -statistics.

Mixing conditions [Rosenblatt, 1956] A stationary sequence { X k } k ∈ Z is called strong mixing , or a-mixing , if α ( m ) → 0 as m → ∞ where α ( m ) = sup | P ( B ∩ A ) − P ( B ) P ( A ) | , (4) A ∈F 0 −∞ , B ∈F ∞ m [Ibragimov, 1962] uniformly strong mixing or ϕ -mixing: ϕ ( m ) → 0 as m → ∞ where | P ( B ∩ A ) − P ( B ) P ( A ) | ϕ ( m ) = sup (5) P ( A ) A ∈F 0 −∞ , B ∈F∞ m , P ( A ) � =0 where F 0 −∞ = σ { X j , j ≤ 0 } and F ∞ m = σ { X j , j ≥ m } . [Ibragimov, Linnik, 1965] If { ξ j } is i.i.d Gaussian sequence then ϕ - mixing is equivalent to the finitness of A 0 := { j ∈ Z : a j � = 0 } .

[Rosenblatt 1980, Andrews 1984] Let ξ j be independent Bernoulli random variables | ρ | < 1. Then the sequence X k = � ∞ j =0 ρ j ξ k − j is not α -mixing. Theorem 1. Let { ξ j ; j ∈ Z } be nondegenerated independent descrete random variables. Moreover, let 0 < δ ≤ ∆ < ∞ be such constants that distance between any two atoms of ξ j is not less than δ and is not greater than ∆ . Finally, let the series (2) be convergent for each k and one of the following two conditions be fulfilled 1) a j = a 0 q | j | if j < 0 and a j = a 0 a j if j ≥ 0 , where a 0 � = 0 , 0 < q < 1 , 0 < a ≤ δ/ (∆ + δ ); 2) 0 < | a k j +1 | ≤ | a k j | δ/ (∆ + δ ) for all j ≥ 0 , where { a k j ; j ≥ 0 } are non-zero coefficients a j ordered by their absolute values. Then the the sequence { X k } does not satisfy α -mixing. If there are only strong inequalities for a and a k j in conditions 1) and 2) then proposition is true when ξ j are arbitrary dependent.

Let { ξ j ; j ∈ Z } be independent bounded random Theorem 2. variables. Let set A − := { j < 0 : a j � = 0 } be infinite and � j ∈ Z | a j | < ∞ . Then the sequence { X k } does not satisfy ϕ -mixing condition. Let { ξ j ; j ∈ Z } be independent bounded random Theorem 3. variables with density p ( x ) , A − be finite and for some C > 0 � R | p ( y + x ) − p ( y ) | dy ≤ C | x | for all x ∈ R . � j | a j | < ∞ , j ∈ Z � | a k 0 | > | a j | for some k 0 . j � = k 0 Then { X k } satisfies ϕ -mixing condition.

Theorem 4. Let { ξ j ; j ∈ Z } be independent nondegenerated random variables, a j > 0 for all j, and the following conditions be fulfilled 1) for some positive constants x 0 , c 0 , c 1 , c 2 , integers j 1 > 0 and j 0 P ( ξ j 0 ≥ x + y ) ≤ c 1 e − c 2 y sup x ≥ x 0 for all y ≥ 0 , P ( ξ j 0 ≥ x ) P ( ξ j ≥ x ) ≤ c 1 e − c 2 x for all x ≥ x 0 , if | j | < j 1 , P ( ξ j ≥ x ) ≤ c 0 P ( ξ j 0 ≥ x ) for all x ≥ x 0 , if | j | ≥ j 1 ; 2) inequality holds a j inf > 0; a j +1 j ∈ Z and one of the following conditions is fulfilled 3) inequalities hold a j +1 � a j ln | j | < ∞ and inf > 0 or a j j ∈ Z j � =0 3 ′ ) for some δ > 0 , c 3 > 0 a j +1 a j | j | δ < ∞ � and ln | j | ≥ c 3 for all j such that | j | is greate enough. a j j ∈ Z Then the corresponding sequence { X k } does not satisfy ϕ -mixing condition.

CLT for functions of moving averages Results for g ( X k ) = h ( { ξ k − j } j ∈ Z ). Let g ( x ) be a Lipschitz function and { ξ j } be i.i.d. Ibragimov, Linnik, 1965, Billingsley, 1968 ∞ � 1 / 2 � � a 2 � < ∞ (10) k n =1 | k |≥ n Ibragimov, Linnik, 1965 � 1+ δ ∞ 2+ δ 2+ δ < ∞ , � E | X 1 | 2+ δ < ∞ , � � � � 1+ δ a k ξ − k (11) E � � � � n =1 | k |≥ n or ∞ � � � � � � | X 1 | < C, a k ξ − k � < ∞ . (12) E � � � n =1 | k |≥ n

Hall, Heyde, 1975. ∞ ∞ � 2 � � � | a k | < ∞ , (13) n =1 | k |≥ n Conditions (10)–(13) imply � | a j | < ∞ . (14) j ∈ Z Conditions (10), (13) are stronger than (14). Also if ξ j are Gaussian then conditions (11), (12) imply (10). Ho, Hsing (1997). a j = 0 for all j < 0, � | a j | < ∞ . j ∈ Z Wu (2002). a j = 0 for all j < 0, � ∞ ∞ � 2 � � a k < ∞ . n =1 k = n

[Dedecker J., Merlevede F., Volny D. (2007)]. If � j ∈ Z | a j | < ∞ , { ξ j } – i.i.d, E ξ 0 = 0, E ξ 2 0 < ∞ , w g ( h, M ) ≤ ChM α , where α ≥ 0, E | ξ 0 | 2+2 α < ∞ , or � ξ 0 � < ∞ , � w g ( c | a k | , � X 0 � ∞ ) < ∞ , k ∈ Z where | g ( x + t ) − g ( x ) | . w g ( h, M ) = sup | t |≤ h, | x |≤ M, | x + t |≤ M Then CLT holds.

[Dedecker J., Merlevede F., Volny D. (2007)]. If a k = 0, k < 0, { ξ j } – i.i.d, E ξ 0 = 0, E ξ 2 0 < ∞ , n � n i =0 | a i | �� a 2 � lim sup i =0 a i | < ∞ , and i = o ( s n ) , | � n n →∞ k =1 i ≥ k where s n = √ n | a 0 + . . . + a n | , g is Lipschitz and g ′ is contnuous then n � 2 1 � σ 2 = E ξ 2 g ( X k ) → d N (0 , σ 2 ) , E g ′ ( X 0 ) � where . 0 s n k =1 [Dedecker J., Merlevede F., Volny D. (2007)]. If a k = 0, k < 0, { ξ j } – i.i.d, E ξ 0 = 0, E ξ 4 0 < ∞ , � ∞ � a 2 E g ′ ( X 0 ) = 0 . � � | a k | � i < ∞ , � k ≥ 0 i = k +1 g ′ is Lipschitz, then CLT holds.

k , C = � | a i | and one of the β l X l Let g ( X k ) := Theorem 5. � l ≥ 0 following conditions ( α ) or ( α 0 ) be fulfilled: ( α ) for some δ > 0  1 2+ δ < ∞ , | β l | · l · C l � E | ξ 0 | (2+ δ ) l �  �     l ≥ 0 ∞ δ 2+ δ < ∞ ,  α ( n ) or �    n =0  ( α 0 ) ξ 0 is bounded and | β l | · l · C l < ∞  �   l ≥ 0  ∞ α ( n ) < ∞ , �    n =0 where α ( n ) is a mixing coefficient for { ξ j } . Then { g ( X k ) } k ≥ 1 satisfies CLT.

Let { ξ j } be i.i.d, β l = 0 and E ξ l 0 = 0 for all odd Theorem 6. numbers l , � 2 � � � | a i a i + k | < ∞ , k � 0 i � l/ 2 � 1 / 2 � � a 2 ( l !) 1 / 2 E | ξ | 2 l � � | β l | 2 < ∞ . i i l � 0 Then { g ( X k ) } k ≥ 1 satisfies CLT.

Limit theorem for U -statistics Second order degenerated (canonical) U -statistics: U n = 1 � f ( X k 1 , X k 2 ) (40) n 1 ≤ k 1 � = k 2 ≤ n where kernel f is degenerated (canonical), i.e. E f ( t, X k ) = E f ( X k , t ) = 0 for all t. (41) Rubin H., Vitale R. A. (1980). { X k } k ≥ 1 are independent. Borisov I. S., Volodko N. V. (2008). { X k } k ≥ 1 are m-dependent or mixing.

Let functions { e i ( t ) : i ≥ 0 } form an orthonormal basis in { h : E h 2 ( X 0 ) < ∞} , e 0 ( t ) ≡ 1. Then E e i ( X 0 ) = 0 for all i ≥ 1, E e 2 i ( X 0 ) = 1 for all i . Functions { e i ( t 1 ) e j ( t 2 ) } i,j ≥ 0 form an orthonormal basis in { h : E h 2 ( X ∗ 0 , X ∗ 1 ) < ∞} . Let E f 2 ( X ∗ 0 , X ∗ 1 ) < ∞ , then � f ( t 1 , t 2 ) = f i 1 ,i 2 e i 1 ( t 1 ) e i 2 ( t 2 ) . (43) i 1 ,i 2 ≥ 1

Theorem 7. Let function f be continuos, functions e i and e i e j be Lipschitz for all i, j ≥ 1, � | a i | < ∞ , (44) i � | f i,j | (1 + Lip ( e i ) Lip ( e j )) < ∞ . (45) i,j ≥ 1 Then d � U n → f i 1 ,i 2 H i 1 ,i 2 ( τ i 1 , τ i 2 ) , (46) i 1 ,i 2 ≥ 1 where { τ j } is a Gaussian sequence of centered random variables with covariations ∞ � σ i,j = cov ( τ i , τ j ) = cov ( e i ( X 0 ) , e j ( X k )) , (47) k = −∞ H i 1 ,i 2 ( τ i 1 , τ i 2 ) = τ i 1 τ i 2 for i 1 � = i 2 , and H i,i ( τ i , τ i ) = τ 2 i − 1.

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