On the LIL for Self-Normalized Sums of IID Random Variables期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

On the LIL for Self-Normalized Sums of IID Random Variables

Authors:	Evarist Giné David M Mason

Abstract:	Let $X,X_i ,i \in \mathbb{N},$ be i.i.d. random variables and let, for each $n \in \mathbb{N},S_n = \sum\nolimits_{i = 1}^n {X_i }$ and $V_n^2 = \sum\nolimits_{i = 1}^n {X_i^2 }$ . It is shown that $\lim \sup _{n \to \infty } {{\|S_n \|} \mathord{\left/ {\vphantom {{\|S_n \|} {(V_n \sqrt {\log \log n} ) < \infty }}} \right. \kern-\nulldelimiterspace} {(V_n \sqrt {\log \log n} ) < \infty }}$ a.s. whenever the sequence of self-normalized sums S _n /V _n is stochastically bounded, and that this limsup is a.s. positive if, in addition, X is in the Feller class. It is also shown that, for X in the Feller class, the sequence of self-normalized sums is stochastically bounded if and only if $\lim \sup _{t \to \infty } {{t\|\mathbb{E}XI(\|X\| \leqslant t)\|} \mathord{\left/ {\vphantom {{t\|\mathbb{E}XI(\|X\| \leqslant t)\|} {\mathbb{E}X^2 I(\|X\| \leqslant t)] < \infty }}} \right. \kern-\nulldelimiterspace} {\mathbb{E}X^2 I(\|X\| \leqslant t)] < \infty }}$

Keywords:	Self-normalized sums law of the iterated logarithm Feller class Student t-statistic
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