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].