Precise lim sup behavior of probabilities of large deviations for sums of i.i.d. random variables

Let { X , X n ; n ≥ 1 } be a sequence of real-valued i.i.d. random variables and let S n = ∑ i = 1 n X i , n ≥ 1 . In this paper, we study the probabilities of large deviations of the form P ( S n > t n 1 / p ) , P ( S n < − t n 1 / p ) , and P ( | S n | > t n 1 / 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 S n / n 1 / 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 lim sup x → ∞ P ( | X | > x 1 / p ) / ϕ ( x ) = 1 , then for every t > 0 , lim sup n → ∞ P ( | S n | > t n 1 / p ) / ( n ϕ ( n ) ) = t p α .