Precise rates in complete moment convergence for ρ-mixing sequences

Abstract

Let X 1 , X 2 , … be a strictly stationary sequence of ρ-mixing random variables with mean zeros and positive, finite variances, set S n = X 1 + ⋯ + X n . Suppose that lim n → ∞ E S n 2 / n = σ 2 > 0 , ∑ n = 1 ∞ ρ 2 / q ( 2 n ) < ∞ , where q > 2 δ + 2 . We prove that, if E X 1 2 ( log + | X 1 | ) δ < ∞ for any 0 < δ ⩽ 1 , then lim ϵ ↓ 0 ϵ 2 δ ∑ n = 2 ∞ ( log n ) δ − 1 n 2 E S n 2 I ( | S n | ⩾ ϵ σ n log n ) = E | N | 2 δ + 2 δ , where N is the standard normal random variable.