Small Deviation Probabilities of Sums of Independent Random Variables

Abstract

Let ξ₁,ξ₂, … be a sequence of independent N(0, l)-distributed random variables (r.v.’s) and let (ø(j))∞_j=₁ be a summable sequence of positive real numbers. The sum S := ∑∞_j=₁ ø(j)ξ² _jis then well defined and one may ask for the small deviation probability of S, i.e. for the asymptotic behavior of ℙ(S < r) as r → 0. In 1974 G. N. Sytaya [S] gave a complete description of this behavior in terms of the Laplace transform of S. Recently, this result was considerably extended to sums S := ∑∞_j=₁ ø(j)Z _j for a large class of i.i.d. r.v.’s Z _j ≥ 0 (cf.[DR], [Lif2]). Yet for concrete sequences (ø(j))∞_j=₁ those descriptions of the asymptotic behavior are very difficult to handle because they use an implicitly defined function of the radius r > 0. In 1986 V. M. Zolotarev [Z2] announced an explicit description of the behavior of ℙ(∑∞_j=₁ ø(j)ξ² _j < r) in the case that ø can be extended to a decreasing and logarithmically convex function on [l,∞). We show that, unfortunately, this result is not valid without further assumptions about the function ∞ (a natural example will be given where an extra oscillating term appears). Our aim is to state and to prove a correct version of Zolotarev’s result in the more general setting of [Lif2], and we show how our description applies in the most important specific examples. For other results related to small deviation problems see [A], [I], [KLL], [Li], [LL], [MWZ], [NS] and [Z1].

Research supported by the DFG-Graduiertenkolleg “Analytische und Stochastische Strukturen und Systeme”, Universität Jena

Research supported by International Science Foundation (ISF) and Russian Foundation for Basic Research (RFBI) and carried out during the author’s sojourn in Strasbourg and Lille-1 universities