Cram'er Type Moderate deviations for the Maximum of Self-normalized Sums

摘要

Let ${ X, X_i , i geq 1}$ be i.i.d. random variables, $S_k$ be the partial sum and $V_n^2 = sum_{1leq ileq n} X_i^2$. Assume that $E(X)=0$ and $E(X^4) < infty$. In this paper we discuss the moderate deviations of the maximum of the self-normalized sums. In particular, we prove that $P(max_{1 leq k leq n} S_k geq x V_n) / (1- Phi(x)) o 2$ uniformly in $x in [0, o(n^{1/6}))$.