A Characterization of Chover-Type Law of Iterated Logarithm

Abstract

Let $0<\alpha \le 2$ and $-\infty <\beta <\infty$. Let $\{X_n;n\ge 1\}$ be a sequence of independent copies of a real-valued random variable $X$ and set $S_n=X_1+\cdots +X_n,n\ge 1$. We say $X$ satisfies the $(\alpha ,\beta )$-Chover-type law of the iterated logarithm (and write $X\in CTLIL(\alpha ,\beta )$) if ${\mathrm{limsup}}_{n\to \infty }{\left|\frac{S_n}{n^{1/\alpha }}\right|}^{(\log \log n{)}^{-1}}=e^{\beta }$ almost surely. This paper is devoted to a characterization of $X\in CTLIL(\alpha ,\beta )$. We obtain sets of necessary and sufficient conditions for $X\in CTLIL(\alpha ,\beta )$ for the five cases: $\alpha =2$ and $0<\beta <\infty$, $\alpha =2$ and $\beta =0$, $1<\alpha <2$ and $-\infty <\beta <\infty$, $\alpha =1$ and $-\infty <\beta <\infty$, and $0<\alpha <1$ and $-\infty <\beta <\infty$. As for the case where $\alpha =2$ and $-\infty <\beta <0$, it is shown that $X\notin CTLIL(2,\beta )$ for any real-valued random variable $X$. As a special case of our results, a simple and precise characterization of the classical Chover law of the iterated logarithm (i.e., $X\in CTLIL(\alpha ,1/\alpha )$) is given; that is, $X\in CTLIL(\alpha ,1/\alpha )$ if and only if $\inf \left\{b:\mathbb{E}\left(\frac{|X{|}^{\alpha }}{(\log (e\vee |X|){)}^{b\alpha }}\right)<\infty \right\}=1/\alpha$ where $\mathbb{E}X=0$ whenever $1<\alpha \le 2$.

11 pages

References in corpus (1)

• Some results on two-sided LIL behavior