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Limit behaviour of sums of independent random variables with respect to the uniform p-adic distribution

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  • Khrennikov, Andrei

Abstract

We investigate (as usual) limit behaviour of sums Sn([omega]) of independent equally distributed random variables. However, limits of probabilities are studied with respect to a p-adic metric (where p is a prime number). We found that (despite of rather unusual features of a p-adic metric) limits of classical probabilities exist in a field of p-adic numbers. These probabilities are rational numbers (which can be calculated by using simple combinatorial considerations). Limit theorems are related to divisibility of sums Sn([omega]) by p. In fact, limits depend on choices of subsequences {Snk([omega])}. We obtain two limit theorems which describe all possible limit behaviours. All considerations are based on one special p-adic probability distribution, namely the uniform distribution.

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  • Khrennikov, Andrei, 2001. "Limit behaviour of sums of independent random variables with respect to the uniform p-adic distribution," Statistics & Probability Letters, Elsevier, vol. 51(3), pages 269-276, February.
    • Handle: RePEc:eee:stapro:v:51:y:2001:i:3:p:269-276
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    1. Khrennikov, Andrew, 1995. "p-adic probability distributions of hidden variables," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 215(4), pages 577-587.
    2. Khrennikov, Andrew, 1998. "p-Adic behaviour of Bernoulli probabilities," Statistics & Probability Letters, Elsevier, vol. 37(4), pages 375-379, March.
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    Cited by:

    1. Kumi Yasuda, 2020. "Large Deviations for Scaled Sums of p-Adic-Valued Rotation-Symmetric Independent and Identically Distributed Random Variables," Journal of Theoretical Probability, Springer, vol. 33(2), pages 1196-1210, June.

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