© Hindawi Publishing Corp. PRECISE LIM SUP BEHAVIOR OF PROBABILITIES OF LARGE DEVIATIONS FOR SUMS OF I.I.D. RANDOM VARIABLES

Let {X,Xn; n ≥ 1} be a sequence of real-valued i.i.d. random variables and let Sn = ∑n i=1Xi, n ≥ 1. In this paper, we study the probabilities of large deviations of the form P(Sn> tn1/p), P(Sn <−tn1/p), and P(|Sn|> tn1/p), where t> 0 and 0<p < 2. We obtain precise asymptotic estimates for these probabilities under mild and easily verifiable conditions. For example, we show that if Sn/n1/p P → 0 and if there exists a nonincreasing positive function φ(x) on [0,∞) which is regularly varying with index α ≤ −1 such that limsupx→∞P(|X |> x1/p)/φ(x) = 1, then for every t> 0, limsupn→∞P(|Sn|> tn1/p)/(nφ(n)) = tpα. 2000 Mathematics Subject Classification: 60F10, 60F05