Small Deviations of Sums of Independent Random Variables T. Dunker, M. A. Lifshits, W. Linde

Let ξ1 , ξ2 , . . . be a sequence of independent N (0, 1)-distributed random variables (r.v.) and let (φ(j))∞ be a summable sequence of positive real numbers. P∞ j=1 Then the sum S := j=1 φ(j)ξj2 is 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. G. N.Sytaya gave in 1974 a complete description of this behavior in terms of the LaplacePtransform of S. Recently, this result was considerably extended to ∞ sums S := j=1 φ(j)Zj for a large class of i.i.d. r.v.’s Zj ≥ 0 by M.A.Lifshits. Yet for concrete sequences (φ(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. InP1986 V. M. Zolotarev announced an explicit description ∞ of the behavior of P ( j=1 φ(j)ξj2 ≤ r) with φ decreasing and logarithmically convex. We show that, unfortunately, this representation is not valid without further assumptions about the function φ (a natural example will be given where an extra oscillating term appears). Next, we state and prove a correct version of Zolotarev’s result in the more general (non-Gaussian) setting of Lifshits’s work. Finally, we show how our representation works in the most important specific examples such as φ(t) = t−A and φ(t) = exp(1 − t).

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