Exact rates in log law for positively associated random variables

Let { X n ; n ⩾ 1 } be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set S n = ∑ k = 1 n X k , M n = max k ⩽ n | S k | , n ⩾ 1 . Suppose σ 2 = E X 1 2 + 2 ∑ k = 2 ∞ E X 1 X k . In this paper, we study the exact convergence rates of a kind of weighted infinite series of P { M n ⩾ ε σ n log n } , P { | S n | ⩾ ε σ n log n } and I { | S n | ⩾ ε σ n log n } as ε ↘ 0 , respectively.