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Large deviations and law of the iterated logarithm for partial sums normalized by the largest absolute observation. (English) Zbl 0869.60025
Let \(X_n, 1\leq n <\infty\), be i.i.d. random variables in the domain of attraction of a stable law with index \(0<\alpha<2\). We also assume that \({\mathbf E}X=0\) if \(1<\alpha<2\), \(X\) is symmetric if \(\alpha=1\). Put \(S_n=\sum_{i=1}^{n}X_i\), \(T_n=S_n/M_n\), where \(M_n=\max_{1\leq i \leq n }|X_i|\). The limit of \(x_n^{-1}\log {\mathbf P}(T_n\geq x_n)\) is found when \(x_n \to \infty\) and \(x_n/n\to 0\) as \(n\to \infty\). The large deviation result is used to prove the law of the iterated logarithm for the self-normalized partial sums \(T_n\).

MSC:
60F10 Large deviations
60F15 Strong limit theorems
60G50 Sums of independent random variables; random walks
60G18 Self-similar stochastic processes
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