Precise rates in complete moment convergence for ρ-mixing sequences

Let X1,X2,…X1,X2,… be a strictly stationary sequence of ρ  -mixing random variables with mean zeros and positive, finite variances, set Sn=X1+⋯+XnSn=X1+⋯+Xn. Suppose that limn→∞ESn2/n=σ2>0, ∑n=1∞ρ2/q(2n)<∞, where q>2δ+2q>2δ+2. We prove that, if EX12(log+|X1|)δ<∞ for any 0<δ⩽10<δ⩽1, thenlimϵ↓0ϵ2δ∑n=2∞(logn)δ−1n2ESn2I(|Sn|⩾ϵσnlogn)=E|N|2δ+2δ, where N is the standard normal random variable.