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

Let X₁, X₂, … 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(Sₙ /Vₙ ≥ x)= (1-Φ(x)) (1+O(1) (1+x)2+δ /dn,δ2+δ) for 0 ≤ x ≤ dn,δ,{1pt} where dn,δ = (∑ᵢ₌₁ⁿ EXᵢ²)1/2/(∑ᵢ₌₁ⁿ E|Xᵢ|2+δ)1/(2+δ) and Vₙ= (∑ᵢ₌₁ⁿ Xᵢ²)1/2. Applications to the Studentized bootstrap and to the self-normalized law of the iterated logarithm are discussed.