On the asymptotic growth rate of the sum of p-adic-valued independent identically distributed random variables

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Abstract

The p-adic semistable laws are characterized as weak limits of scaled summands of p-adic-valued, rotation-symmetric, independent, and identically distributed random variables whose tail probability satisfies some condition. In this article it is verified such scaled sums do not converge in probability, and some more precise estimates, corresponding to the law of iterated logarithm in the real-valued setting, are given to the asymptotic growth rate of the sum. The critical scaling order is explicitly given, over which the scaled sum almost surely converges to 0. On the other hand, under the critical order, the limit superior of the p-adic norm of the scaled sum diverges almost surely. Furthermore it is shown that, at the critical order, a crucial change of the asymptotic behavior of the scaled sum occurs according to the decay of the tail probability of the random variables. In this situation, the critical value for the order of the tail probability is also found. [MediaObject not available: see fulltext.]

Original languageEnglish
Pages (from-to)449-460
Number of pages12
JournalArabian Journal of Mathematics
Volume3
Issue number4
DOIs
Publication statusPublished - 2014 Dec 1

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P-adic
Identically distributed
Random variable
Tail Probability
Converge
Weak Limit
Law of the Iterated Logarithm
Diverge
Critical value
Asymptotic Behavior
Scaling
Decay
Norm
Estimate

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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abstract = "The p-adic semistable laws are characterized as weak limits of scaled summands of p-adic-valued, rotation-symmetric, independent, and identically distributed random variables whose tail probability satisfies some condition. In this article it is verified such scaled sums do not converge in probability, and some more precise estimates, corresponding to the law of iterated logarithm in the real-valued setting, are given to the asymptotic growth rate of the sum. The critical scaling order is explicitly given, over which the scaled sum almost surely converges to 0. On the other hand, under the critical order, the limit superior of the p-adic norm of the scaled sum diverges almost surely. Furthermore it is shown that, at the critical order, a crucial change of the asymptotic behavior of the scaled sum occurs according to the decay of the tail probability of the random variables. In this situation, the critical value for the order of the tail probability is also found. [MediaObject not available: see fulltext.]",
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