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

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