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Precise asymptotics in the law of the iterated logarithm. (English) Zbl 1044.60024

Summary: Let \(X,X_1,X_2,\dots\) be i.i.d. random variables with mean 0 and positive, finite variance \(\sigma^2\), and set \(S_n=X_1+ \cdots+X_n\), \(n>1\). Continuing earlier work related to strong laws, we prove the following analogs for the law of the iterated logarithm: \[ \lim_ {\varepsilon\searrow\sigma\sqrt 2}\sqrt{\varepsilon^2-2\sigma^2} \sum_{n\geq 3} \frac 1n P\bigl(| S_n|\geq \varepsilon\sqrt {n \log \log n}+a_n\bigr)= \sigma\sqrt 2 \] whenever \(a_n=O(\sqrt n(\log\log n)^{-\gamma})\) for some \(\gamma\geq 1/2\) (assuming slightly more than finite variance), and \[ \lim_{\varepsilon\searrow 0}\varepsilon^2 \sum_{n\geq 3} \frac{1} {n\log n}P\bigl(| S_n|\geq \varepsilon \sqrt {n\log\log n}\bigr) =\sigma^2. \]

MSC:

60F15 Strong limit theorems
60G50 Sums of independent random variables; random walks
60E15 Inequalities; stochastic orderings
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