Some results on convergence rates for probabilities of moderate deviations for sums of random variables

Let X , X n , n ≥ 1 be a sequence of iid real random variables, and S n = ∑ k = 1 n X k , n ≥ 1 . Convergence rates of moderate deviations are derived, i.e., the rate of convergence to zero of certain tail probabilities of the partial sums are determined. For example, we obtain equivalent conditions for the convergence of series ∑ n ≥ 1 ( ψ 2 ( n ) / n ) P ( | S n | ≥ n φ ( n ) ) only under the assumptions convergence that E X = 0 and E X 2 = 1 , where φ and ψ are taken from a broad class of functions. These results generalize and improve some recent results of Li (1991) and Gafurov (1982) and some previous work of Davis (1968). For b ∈ [ 0 , 1 ] and ϵ > 0 , let λ ϵ , b = ∑ n ≥ 3 ( ( log log n ) b / n ) I ( | S n | ≥ ( 2 + ϵ ) n log log n ) . The behaviour of E λ ϵ , b as ϵ ↓ 0 is also studied.