Precise asymptotics in the deviation probability series of self-normalized sums

Abstract

Let X , X 1 , X 2 , … be a sequence of nondegenerate i.i.d. random variables with zero means. Set S n = X 1 + ⋯ + X n and W n 2 = X 1 2 + ⋯ + X n 2 . In the present paper we examine the precise asymptotic behavior for the general deviation probabilities of self-normalized sums, S n / W n . For positive functions g ( x ) , ϕ ( x ) , α ( x ) and κ ( x ) , we obtain the precise asymptotics for the following deviation probabilities of self-normalized sums: α ( ϵ ) ∑ n = 1 ∞ g ( ϕ ( n ) ) ϕ ′ ( n ) E [ ( S n / W n ) 2 I ( | S n | ⩾ W n ( ϵ ϕ ( n ) + κ ( n ) ) ) ] . The results given can be considered as the generalization of that in the complete moment convergence, law of iterated logarithm and large deviation for self-normalized sums.