A limit result for self-normalized random sums

Abstract

Suppose {X, X_n;n≥1} is a sequence i.i.d.r.v. withEX=0 andEX²<∞. Shao (1995) proved a conjecture of Révész (1990); ifP(X=±1)=1/2, then

$$\mathop {lim}\limits_{n \to \infty } \mathop {max}\limits_{0 \leqslant j< n} \mathop {max}\limits_{1 \leqslant k \leqslant n - j} \frac{{\sum\limits_{i = j + 1}^{i = j + k} {X_i } }}{{(2klogn)^{1/2} }} = 1 a. s.$$

. Furthermore he conjectured that

$$1 \leqslant \mathop {lim}\limits_{n \to \infty } \mathop {max}\limits_{0 \leqslant j< n} \mathop {max}\limits_{1 \leqslant k \leqslant n - j} \frac{{\sum\limits_{i = j + 1}^{i = j + k} {X_i } }}{{\{ \sum\limits_{i = j + 1}^{i = j + k} {X_i^2 (2klogn)} \} ^{1/2} }} = {\rm K}< \infty a. s.$$

. In this paper we prove that if$\mathop {sup}\limits_{b > 0} P(X = b) \geqslant P(X = 0)$ then this conjecture is ture.