Convergence Rates for Probabilities of Moderate Deviations for Multidimensionally Indexed Random Variables

Abstract

Let{X,Xn¯;n¯∈Z+d}be a sequence of i.i.d. real-valued random variables, andSn¯=∑k¯≤n¯Xk¯,n¯∈Z+d. Convergence rates of moderate deviations are derived; that is, the rates of convergence to zero of certain tail probabilities of the partial sums are determined. For example, we obtain equivalent conditions for the convergence of the series∑n¯b(n¯)ψ2(a(n¯))P{|Sn¯|≥a(n¯)ϕ(a(n¯))}, wherea(n¯)=n11/α1⋯nd1/αd,b(n¯)=n1β1⋯ndβd,ϕandψare taken from a broad class of functions. These results generalize and improve some results of Li et al. (1992) and some previous work of Gut (1980).