Precise asymptotics in the law of the iterated logarithm

Abstract

Let $ X, X_1, X_2 \ldots$ be i.i.d. random variables with mean 0 and positive, finite variance $\sigma^2$, and set $S_n = X_1 + \cdots + X_n, n \geq 1$. Continuing earlier work related to strong laws, we prove the following analogs for the law of the iterated logarithm: $$\lim_{\varepsilon\searrow\sigma\sqrt{2}}\sqrt{\varepsilon^2−2\sigma^2}\sum_{n\ge3}\frac{1}{n}P(|S_n|\ge\varepsilon\sqrt{n\log\log n}+a_n)=\sigma\sqrt{2}$$ whenever $a_n = O(\sqrt{n}(\log \log n)^{-\gamma})$ for some $\gamma \geq 1/2$ (assuming slightly more than finite variance), and $$\lim_{\varepsilon\searrow 0}\varepsilon^2\sum_{n\ge3}\frac{1}{n \log n}P(|S_n|\ge\varepsilon\sqrt{n\log\log n})=\sigma^{2}.$$