A Characterization of Chover-Type Law of Iterated Logarithm

Let $0 < α\leq 2$ and $- \infty < β< \infty$. Let $\{X_{n}; n \geq 1 \}$ be a sequence of independent copies of a real-valued random variable $X$ and set $S_{n} = X_{1} + \cdots + X_{n}, ~n \geq 1$. We say $X$ satisfies the $(α, β)$-Chover-type law of the iterated logarithm (and write $X \in CTLIL(α, β)$) if $\limsup_{n \rightarrow \infty} \left| \frac{S_{n}}{n^{1/α}} \right|^{(\log \log n)^{-1}} = e^β$ almost surely. This paper is devoted to a characterization of $X \in CTLIL(α, β)$. We obtain sets of necessary and sufficient conditions for $X \in CTLIL(α, β)$ for the five cases: $α= 2$ and $0 < β< \infty$, $α= 2$ and $β= 0$, $1 < α< 2$ and $-\infty < β< \infty$, $α= 1$ and $- \infty < β< \infty$, and $0 < α< 1$ and $-\infty < β< \infty$. As for the case where $α= 2$ and $-\infty < β< 0$, it is shown that $X \notin CTLIL(2, β)$ 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., $X \in CTLIL(α, 1/α)$) is given; that is, $X \in CTLIL(α, 1/α)$ if and only if $\inf \left \{b:~ \mathbb{E} \left(\frac{|X|^α}{(\log (e \vee |X|))^{bα}} \right) < \infty \right\} = 1/α$ where $\mathbb{E}X = 0$ whenever $1 < α\leq 2$.

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