Some Results on Small-Deviation Probability Convergence Rates for Sums of Independent Random Variables

Abstract

Let ﹛Xj,j = 1, 2, 3, …﹜ be a sequence of independent, non-degenerate random variables and writeUnder quite a diverse variety of conditions we may obtainas n → ∞ for all x, — ∞ < x < ∞, and some real p ⩾ 0. For example, suppose the ﹛Xj﹜ happen to be distributed identically and belong to the domain of normal attraction of a symmetric stable law with characteristic exponent α, 0 < α ⩽ < 2, α ≠ 1.