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Precise Asymptotics in Spitzer's Law of Large Numbers

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Abstract

Let X, X 1, X 2,... be a sequence of i.i.d. random variables such that EX=0, assume the distribution of X is attracted to a stable distribution with exponent α<1, and set S n=X 1+ ··· +X n. We prove that

$$\sum\limits_{n{\text{ }} \geqslant {\text{ }}1} {{\text{ }}\frac{{\text{1}}}{{\text{n}}}{\text{ }}} P(|S_n | \geqslant \varepsilon n){\text{ }} \sim {\text{ }}\frac{\alpha }{{\alpha - 1}}{\text{ }}( - \log \varepsilon {\text{) as }}\varepsilon \searrow {\text{0}}$$

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  1. Centre of Mathematical Statistics, Romanian Academy, Calea 13 Septembrie, nr. 13, 76100, Bucharest 5, Romania

    Aurel Spătaru

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  1. Aurel Spătaru
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Spătaru, A. Precise Asymptotics in Spitzer's Law of Large Numbers. Journal of Theoretical Probability 12, 811–819 (1999). https://doi.org/10.1023/A:1021636117551

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