Abstract
Let {X n , n ≥ 1} be a strictly stationary ρ--mixing sequence of positive random variables with EX1 = μ > 0 and Var(X1) = σ2 < ∞. Denote and the coefficient of variation. Under suitable conditions, by the central limit theorem of weighted sums and the moment inequality we show that where with is the distribution function of the random variable , and is a standard normal random variable. MR(2000) Subject Classification: 60F15.
Highlights
Introduction and main resultsFor a random variable X, define ∥X∥p = (E|X|p)1/p
Let C be a class of functions which are coordinatewise increasing
R--mixing random variables include negatively associated (NA) and r*-mixing random variables, which have a lot of applications, their limit properties have aroused wide interest recently, and a lot of results have been obtained, such as the weak convergence theorems, the central limit theorems of random fields, Rosenthal-type moment inequality, see [1,2,3,4]
Summary
A sequence {Xn, n ≥ 1} is called r*-mixing if ρ ∗ (s) = sup ρ (S, T) ; S, T ⊂ N, dist(S, T) ≥ s → 0 as s → ∞, where ρ(S, T) = sup E(f − Ef )(g − Eg)/ f − Ef 2 · g − Eg 2 ; f ∈ L2(σ (S)), g ∈ L2(σ (T)). A sequence {Xn, n ≥ 1} is called r--mixing, if ρ (s) = sup ρ (S, T); S, T ⊂ N, dist(S, T) ≥ s → 0 as s → ∞. R--mixing random variables include NA and r*-mixing random variables, which have a lot of applications, their limit properties have aroused wide interest recently, and a lot of results have been obtained, such as the weak convergence theorems, the central limit theorems of random fields, Rosenthal-type moment inequality, see [1,2,3,4]. Zhou [5] studied the almost sure central limit theorem of r--mixing sequences by the conditions provided by Shao: on the conditions of central limit theorem, and if ε0 > 0, Var n1 f i=1 i
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