Abstract
Let { X n ,n≥1} be a sequence of strictly stationary ϕ-mixing positive random variables which are in the domain of attraction of the normal law with E X 1 =μ>0, possibly infinite variance and mixing coefficient rates ϕ(n) satisfying ∑ n ≥ 1 ϕ 1 / 2 ( 2 n )<∞. Under suitable conditions, we here give an almost sure central limit theorem for self-normalized products of partial sums, i.e., lim n → ∞ 1 D n ∑ m = 1 n d m I ( ( ∏ k = 1 m S k k μ ) μ / ( β V m ) ≤ x ) =F(x)a.s. for any x∈R, where F is the distribution function of the random variable e 2 N and N is a standard normal random variable.MSC:60F15.
Highlights
Introduction and main resultsThe almost sure central limit theorem (ASCLT) was first introduced independently by Brosamler [ ] and Schatte [ ]
It is known that the class of sequences satisfying the ASCLT is larger than the class of sequences satisfying the central limit theorem
We refer to Gonchigdanzan and Rempala [ ] on the ASCLT for the products of partial sums, Gonchigdanzan [ ] on the ASCLT for the products of partial sums with stable distribution
Summary
In this paper we study the almost sure central limit theorem, containing the general weight sequences, for weakly dependent random variables. Let {Xn, n ≥ } be a sequence of strictly stationary φ-mixing positive random variables which are in the domain of attraction of the normal law with EX = μ > , possibly infinite variance and mixing coefficients φ(n) satisfying n≥ φ / ( n) < ∞. Let {Xn, n ≥ } be a strictly stationary φ-mixing sequence of positive random variables such that EX = μ > , Var(X ) = σ < ∞ and n≥ φ / ( n) < ∞. If {Xn, n ≥ } is a sequence of i.i.d. positive random variables such that EX = μ > and X belongs to the domain of attraction of the normal law, Theorem . If the assumptions of Theorem . hold and there exists a positive constant such that n
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