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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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