Báo cáo nghiên cứu khoa học: "Mở rộng một số định lí giới hạn cho các biến ngẫu nhiên phụ thuộc âm và phụ thuộc âm tuyến tính" potx

1 1. X 1 , , X n P ( n  i=1 [X i  x i ])  n  i=1 P (X i  x i ), ∀x 1 , , x n ∈ R (1) P ( n  i=1 [X i > x i ])  n  i=1 P (X i > x i ), ∀x 1 , , x n ∈ R. (2) X 1 , , X n , X 1 , , X n , i = j X i , X j (Ω, F, P ) Ω = {1, 2, 3, 4} F = {∀A : A ⊂ Ω} P (A) = |A| 4 A = {1, 2} B = {2, 3, 4} I A I B 1 2. X 1 , , X n , A, B (λ k , k ∈ A), (λ l , l ∈ B)  k∈A λ k X k ,  l∈B λ l X l 2 1 X 1 , , X n f 1 , , f n f(X 1 ), , f(X n ) 2 X 1 , X 2 EX 1 X 2  EX 1 EX 2 cov(X 1 , X 2 )  0. 1 X 1 , , X n D(X 1 + + X n )  DX 1 + + DX n X 1 , , X n i = j cov(X i , X j )  0. D( n  k=1 X k ) = n  k=1 D(X k ) +  i<j cov(X i , X j )  n  k=1 D(X k ). 1 {X n , n ≥ 1} 1 n 2 n  i=1 DX i → 0 khi n → ∞ {X n } 1 n  n  i=1 X i − n  i=1 EX i  P −→ 0 khi n → ∞. ε > 0 P  | 1 n n  i=1 X i − 1 n n  i=1 EX i | ≥ ε   D( 1 n n  i=1 X i ) ε 2 = D( n  i=1 X i ) n 2 ε 2  n  i=1 DX i n 2 ε 2 . 1 n 2 n  i=1 DX i → 0 khi n → ∞ lim n→∞ P  | 1 n n  i=1 X i − 1 n n  i=1 EX i | ≥ ε  = 0. 1 n  n  i=1 X i − n  i=1 EX i  P −→ 0 khi n → ∞. 2 {X n , n ≥ 1} C > 0 DX n  C n ≥ 1 {X n } 3 {X n , n ≥ 1} EX 1 = a DX 1 = σ 2 n  i=1 X i P −→ a khi n → ∞. 3 X 1 , , X n |Φ(r 1 , , r m ) − m  j=1 Φ j (r j )|  m  k,l=1 |r k r l cov(X k , X l )| Φ(r 1 , , r m ) = E(exp[(i m  j=1 r j X j ]), Φ j (r j ) = E(exp[ir j X j ]). 4 (F n ) (ϕ n ) F n w −→ F ϕ n → ϕ ϕ F 5 X n w −→ C = const X n P −→ C |a 1 a 2 a n − b 1 b 2 b n |  n  k=1 |a k − b k |, |a k |  1, |b k |  1; (3) |e itx − 1 − itx|  2h 1 (t)g 1 (x); (4) |e itx − 1 − itx + t 2 x 2 2 |  h 2 (t)g 2 (x), (5) h 1 (t) = max(|t|, t 2 ) g 1 (x) = min(|x|, x 2 ), h 2 (t) = max(t 2 , |t| 3 ), g 2 (x) = min(x 2 , |x| 3 ) x, t ∈ R 2 {X ni , 1  i  n, n ≥ 1} EX ni = 0,  i<j cov(X ni , X nj ) n→∞ −−−→ 0 M n = n  i=1 E min(|X ni |, |X ni | r ) n→∞ −−−→ 0 r ∈ (1; 2) S n = n  i=1 X ni P −→ 0 | n  k=1 Ee itX nk − 1| = | n  k=1 Ee itX nk − n  k=1 1|  n  k=1 |Ee itX nk − 1| = = n  k=1 |E(e itX nk − 1 − itX nk )|  n  k=1 2h 1 (t)Eg 1 (X nk ) = 2h 1 (t) n  k=1 E min(|X nk |, X 2 nk ). (6) 1 < r  2 min(|x|, x 2 )  min(|x|, |x| r ) 0  n  k=1 E min(|X nk |, X 2 nk )  n  k=1 min(|X nk |, |X nk | r ) n→∞ −−−→ 0. Suy ra n  k=1 E min(|X nk |, X 2 nk ) n→∞ −−−→ 0. (7) n  k=1 Ee itX nk n→∞ −−−→ 1. (8) |E exp(i n  k=1 X nk ) − n  k=1 E exp(iX nk )|   kl cov(X nk , X nl ) n→∞ −−−→ 0. (9) E exp(i n  k=1 X nk ) → 1, hay ϕ s n → 1. (10) 1 = e it0 X = 0 S n w −→ 0. (11) S n P −→ 0. 4 {X nk , k = 1, , n, n ≥ 1} EX nk = 0  i<j cov(X ni , X nj ) n→∞ −−−→ 0 n  k=1 E|X nk |  C < ∞ L 1 n (ε) = n  k=1 E(|X nk |I(|X nk | > ε)) n→∞ −−−→ 0 ε > 0 S n P −→ 0. 0 < ε < 1 M n  n  k=1 E(|X nk |I(|X nk | > ε)) + n  k=1 E(|X nk | r I(|X nk |  ε))  L 1 n (ε) + ε r−1 C. M n n→∞ −−−→ 0 5 (X k ) EX 1 = a EX 2 1 = C < ∞ E(|X 1 − a|I(|X 1 − a| > εn)) n→∞ −−−→ 0 ε > 0 X 1 + +X n n P −→ a. X nk = X k −a n , k  n {X nk , 1  k  n, n ≥ 1} EX nk = 0,  i<j cov(X ni , X nj ) < C n n→∞ −−−→ 0 n  k=1 E|X k n | = n  k=1 E| X k − a n | = E|X 1 − a| < ∞. L 1 n (ε) = n  k=1 E(|X nk |I(|X nk | > ε)) = n  k=1 E(| X k − a n |I(| X k − a n | > ε)) = = E(|X 1 − a|I(|X 1 − a| > εn)) n→∞ −−−→ 0. 6 {X nk , k = 1, , n, n ≥ 1} EX nk = 0,  i<j cov(X ni , X nj ) → 0 n  k=1 E|X nk | r → 0 r ∈ [1; 2] S n P −→ 0. 3 {X nk , 1  k  n, n ≥ 1}          EX nk = 0, k = 1, , n n  k=1 DX nk = 1,  i<j cov(X ni , X nj ) n→∞ −−−→ 0 i, j = 1, , n. (12) S n = n  k=1 X nk , σ 2 nk = DX nk M 2 n = n  k=1 E min(|X nk | 2 , |X nk | s ) n→∞ −−−→ 0, (13) s ∈ [2; 3] (S n ) N(0, 1). ϕ S n (t) → e − t 2 2 , ∀t ∈ R |ϕ S n (t) − n  k=1 ϕ X nk (t)|   i<j cov(X ni , X nj ) n→∞ −−−→ 0 ϕ S n (t) n→∞ −−−→ n  k=1 ϕ X nk (t) n  k=1 ϕ X nk (t) n→∞ −−−→ e − t 2 2 , t ∈ R | n  k=1 ϕ X nk (t) − e − t 2 2 | = | n  k=1 ϕ X nk (t) − n  k=1 e − t 2 σ 2 nk 2 |  n  k=1 |ϕ X nk (t) − e − t 2 σ 2 nk 2 |   n  k=1 |E(e itX nk − 1 − itX nk + t 2 X 2 nk 2 ) + n  k=1 |e − t 2 σ 2 nk 2 − 1 + t 2 σ 2 nk 2 |   h 2 (t) n  k=1 min(X 2 nk , |X nk | 3 ) + t 4 8 n  k=1 σ 2 nk   h 2 (t) n  k=1 min(X 2 nk , |X nk | s ) + t 4 8 max kn σ 2 nk . (14) 0 < ε < 1 σ 2 nk = E(X 2 nk I(|X nk |  ε)) + E(X 2 nk I(|X nk | > ε)  ε 2 + E(X 2 nk I(|X nk | > 1)) + E(X 2 nk I(ε < |X nk |  1))  ε 2 + E(X 2 nk I(|X nk | > 1)) + 1 ε s−2 E(X s nk I(1 ≥ |X nk | > ε))  ε 2 + 1 ε s−2 E min(X 2 nk , |X nk | s ). (15) 7 {X nk , 1  k  n, n ≥ 1} EX ni = 0 S‘ 2 n = D( n  i=1 X ni ) n→∞ −−−→ ∞, 1 S‘ 2 n  i<j cov(X ni , X nj ) n→∞ −−−→ 0 n  i=1 E(X 2 ni I(|X ni | ≥ εS‘ 2 n )) = 0(S‘ 2 n ) S‘ −1 n = n  i=1 X ni N(0, 1). S 2 n = n  i=1 DX nk {X nk , 1  k  n, n ≥ 1} S 2 n = D( n  k=1 X nk ) = n  k=1 D(X nk ) +  i<j cov(X ni , X nj )  n  k=1 D(X nk ) = S 2 n . 1 S 2 n  i<j cov(X ni , X nj ) n→∞ −−−→ 0; (16) n  i=1 E(X 2 ni I(|X ni | > ε  S 2 n )) = 0(S 2 n ). (17) lim n→∞ S‘ 2 n S 2 n = lim n→∞ ( S 2 n S 2 n + 1 S 2 n  i<j cov(X ni , X nj )) = 1 + lim n→∞ 1 S 2 n  i<j cov(X ni , X nj ) = 1.  S 2 n −1 n  i=1 X ni d −→ X X N(0, 1) Z nk = X nk √ S 2 n {Z nk , 1  k  n, n ≥ 1} ε > 0 n  k=1 EX 2 nk = n  k=1 E(Z 2 nk I(|Z nk |  ε)) + n  k=1 E(Z 2 nk I(|Z nk | > ε))  ε 2 + n  k=1 1 S 2 n EX 2 nk I(|X nk | > ε  S 2 n ). (18) n  k=1 EZ 2 nk n→∞ −−−→ 0 {Z nk , 1  k  n, n ≥ 1} n  k=1 Z nk d −→ X X N(0.1)