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

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 Cramer-type large deviation result for the standardized partial sums. In this paper, we show that a Cramer-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 2+δ n,δ ) for 0 ≤ x ≤ d n,δ , where d n,δ = (Σ n i=1 EX 2 i ) 1/2 /(Σ n i=1 E|X i | 2+δ ) 1/(2+δ) and V n = (Σ n i=1 X 2 i ) 1/2 . Applications to the Studentized bootstrap and to the self-normalized law of the iterated logarithm are discussed.