Precise rates in the law of the iterated logarithm under dependence assumptions (Q928235)

Let \(X_1,X_2,\dots\) be i.i.d random variables with mean \(0\), variance \(\sigma^2\) and let \(S_n\), \(n\geq 1\), denote the partial sums. Spătaru and the reviewer have proved the following result related to the law of the iterated logarithm: If \(EX^2(\log^+\log^+|X|)^{1+ \delta}<\infty\) for some \(\delta> 0\), then \[ \lim_{\varepsilon\searrow 0}\, \sum_{n\geq 3} {1\over n} P(|S_n|> \varepsilon \sqrt{\sigma^2 n\log\log n}+ a_n)= \sqrt{2}, \] for \(a_n={\mathcal O}(\sqrt{n}/(\log\log n)^\gamma)\), where \(\gamma> 1/2\). The authors of the present paper prove the analog under certain mixing conditions on the summands as well as som other related results.