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

Abstract

LetX,Xn,n≥1be a sequence ofiidreal random variables, andSn=∑k=1nXk,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(ψ2(n)/n)P(|Sn|≥nφ(n))only under the assumptions convergence thatEX=0andEX2=1, whereφandψare 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). Forb∈[0,1]andϵ>0, letλϵ,b=∑n≥3((loglogn)b/n)I(|Sn|≥(2+ϵ)nloglogn).The behaviour ofEλϵ,basϵ↓0is also studied.