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

Let {X,Xn;n ∈ Z } be a sequence of i.i.d. real-valued random variables, and Sn ∑ k≤n Xk , n ∈ Z . 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 ψ 2 a n P{|Sn| ≥ a n φ a n }, where a n n1 1 · · ·nd d , b n n β1 1 · · ·n βd d , φ and ψ are 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 .