On the Self-Normalized Cramér-type Large Deviation

Abstract

For the self-normalized sum, $$S_n/V_n$$ , it is shown that $$P(S_n/V_n\ge x)/(1-\Phi(x))$$ converges to 1, uniformly in a region, under the optimal assumption that the sampled distribution is in the domain of attraction of the normal law. Bounds for this convergence are given and their applications to exponential non-uniform Berry–Esseen bound are also discussed.