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A tauberian theorem for random walk

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Abstract

LetX 1,X 2,... be independent random variables, all with the same distribution symmetric about 0;

$$S_n = \sum\limits_{i = 1}^n {X_i } $$

It is shown that if for some fixed intervalI, constant 1<a≦2 and slowly varying functionM one has

$$\sum\limits_{k = 1}^n {P\{ S_k \in I\} \sim \frac{{n^{1 - 1/\alpha } }}{{M(n)}}} (n \to \infty )$$

then theX i belong to the domain of attraction of a symmetric stable law.

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Authors and Affiliations

  1. Aarhus University, Aarhus, Denmark

    Harry Kesten

  2. Cornell University, Ithaca, New York

    Harry Kesten

Authors
  1. Harry Kesten
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Additional information

Research supported by the National Science Foundation under grant GP 7128.

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Kesten, H. A tauberian theorem for random walk. Israel J. Math. 6, 279–294 (1968). https://doi.org/10.1007/BF02760260

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  • DOI: https://doi.org/10.1007/BF02760260

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