Báo cáo toán học: 'Renewal Process for a Sequence of Dependent Random Variables'

Chúng tôi điều tra một quá trình đổi mới N (t) = max {n ≥ 1: Sn = n i = 1 Xi ≤ t} cho t ≥ 0, nơi X1, X2, ... với P (Xi ≥ 0) = 1 (i = 1, 2, ...) là một chuỗi các biến ngẫu nhiên mdependent hoặc trộn. Chúng tôi cung cấp một điều kiện mà theo đó N (t) có thời điểm hữu hạn. Mạnh mẽ pháp luật của một số lượng lớn và các định lý giới hạn trung tâm cho các chức năng N (t) được đưa ra. 1. Chuẩn bị và ký hiệu...
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Báo cáo toán học: "Renewal Process for a Sequence of Dependent Random Variables" 9LHWQDP -RXUQDO Vietnam Journal of Mathematics 33:1 (2005) 73–83 RI 0$7+(0$7,&6 ‹ 9$67  Renewal Process for a Sequence of Dependent Random Variables Bui Khoi Dam Hanoi Institute of Mathematics, 18 Hoang Quoc Viet Road, 1037 Hanoi, Vietnam Received October 15, 2003 Revised April 25, 2004 n Abstract. We investigate a renewal process N (t) = max{n ≥ 1 : Sn = X i ≤ t} i=1 for t ≥ 0 where X1 , X2 , ... with P (Xi ≥ 0) = 1 (i = 1, 2, ... ) is a sequence of m- dependent or mixing random variables. We give such a condition under which N (t) has finite moment. Strong law of large numbers and central limit theorems for the function N (t) are given. 1. Preliminaries and Notations Let (Ω, A, P ) be a probability space and let X0 , X1 , X2 , ... be non negative ran- n dom variables with P (X0 = 0) = 1, Sn = Xi . It is well known that if the i=1 sequence X1 , X2 , ... is independent and identically distributed, then the counting n process N (t) = max{n ≥ 1 : Sn = Xi ≤ t}, t ≥ 0 is called a renewal process. i=1 In this article, we investigate generalized renewal process, i.e. we suppose that our basic sequence X0 , X1 , X2 , ... is a sequence of m - independent or mixing radom variables. We denote Fn = σ (X0 , X1 , ..., Xn ), F k = σ (Xk , Xk+1 , ...). Now we begin this section with some definitions. Definition 1.1. A sequence of random variables (Xn )n≥0 is called m-dependent if the sigma-fields Fn and F n+k are independent for all k > m. Definition 1.2. We consider the following quantities 74 Bui Khoi Dam α(n) = sup{|P (A.B ) − P (A).P (B )| : A ∈ Fk , B ∈ F k+n }; ρ(n) = sup{|Cov (X.Y )|/(V (X )1/2 .V (Y )1/2 ) : X ∈ Fk , Y ∈ F k+n }; φ(n) = sup{|P (B |A) − P (B )| : A ∈ Fk , P (A) > 0; B ∈ F k+n }. A sequence of random variables (Xn )n≥0 is said to be α-mixing (resp. ρ-mixing, φ-mixing) if lim α(n) = 0 (resp. lim ρ(n) = 0, lim φ(n) = 0). n→+∞ n→+∞ n→+∞ 2. Results Theorem 2.1. Let (Xn )n≥0 be a sequence of nonnegative random variables. Denote pi = P (Xi ≥ a) where a is a positive constant and N (t) = max{n ≥ 1 : n Xi ≤ t}. Suppose that either Sn = 1=1 (i) (Xn )n≥0 is a (m − 1)- dependent random variables, (m ≤ 1) such that n pr+im ≥ Ar .nαr , 0 < Ar < +∞, αr > 0 for all n ≥ 1, m − 1 ≥ r ≥ 0, i=1 or (ii) (Xn )n≥0 is a φ-mixing sequence of random variables such that lim inf pn = p > 0. Then E [N (t)]l < +∞, ∀l. We need the following lemma to prove the theorem. Lemma 2.1. Let (Xn )n≥0 be a sequence of non negative, independent random variables such that n pi ≥ A.nα , 0 < A < +∞, α > 0 for all n ≥ 1. i=1 Then E [N (t)]l < +∞, ∀l. Proof. From the definition of N (t), it is easy to see that N (t) is a non decreasing ¯ function in t. We define new random variables Xn as follows: for a given positive number a, we put ¯ n ≥ 1, Xn = 1(a,∞) (Xn ), n ¯ ¯ Sn = Xi , i=1 and N (t) = max{n ≥ 1 : Sn ≤ t}. ¯ ¯ It is easy to see that 0 ≤ N (t) ≤ N (t/a) for all t > 0. ¯ Renewal Process for a Sequence of Dependent Random Variables 75 ¯ This guarantees that , we can investigate the function N (t) instead of the func- tion N (t). ¯ ¯ ¯ ¯ P (N (j ) = n) = P (X1 + X2 + ... + Xn = j ). J Denote by In (1 ≤ j ≤ n) the set of all combinations of j numbers from the set j {1, 2, ..., n}. For i1 , i2 , ..., ij ∈ In , we consider the following events: A{i1 , i2 , ..., in } = {Xi1 = · · · = Xij = 1}, ¯ ¯ A{i1 , i2 , ..., in } = {Xi(1) = · · · = Xi(n−j ) = 0}, ¯ ¯ ¯ where {i(1), ..., i(n − j )} is the complement of {i1 , ..., ij }, i.e. {i(1), ..., i(n − j )} = {1, 2, ..., n} {i1 , ..., ij }. We obtain the following relations { X1 + X2 + · · · + Xn = j } = ¯ ¯ ¯ A{i1 , ..., ij } ∩ A{i1 , ..., ij } ¯ ...