Self-normalized cramér-type large deviations for independent random variables

Abstract

Let X 1, X 2, . . . be independent random variables with zero means and finite variances. It is well known that a finite exponential moment assumption is necessary for a Cramér-type large deviation result for the standardized partial sums. In this paper, we show that a Cramér-type large deviation theorem holds for self-normalized sums only under a finite (2 + δ)th moment, 0 < δ ≤ 1. In particular, we show P(S n/V n ≥ x) = (1 - Φ(x))(1 + O(1)(1 + x) 2+δ/d n,δ 2+δ) for 0 ≤ x ≤ d n,δ, where d n,δ = (∑ i=1 n EX i 2) 1/2/(∑ i=1 n EX i 2+δ) 1/(2+δ) and V n = (∑ i=1 n X i 2)1/2. Applications to the Studentized bootstrap and to the self-normalized law of the iterated logarithm are discussed.