TY - JOUR

T1 - On the asymptotic growth rate of the sum of p-adic-valued independent identically distributed random variables

AU - Yasuda, Kumi

N1 - Publisher Copyright:
© 2014, The Author(s).
Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2014/12/1

Y1 - 2014/12/1

N2 - 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.]

AB - 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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U2 - 10.1007/s40065-014-0106-5

DO - 10.1007/s40065-014-0106-5

M3 - Article

AN - SCOPUS:85012187599

SN - 2193-5343

VL - 3

SP - 449

EP - 460

JO - Arabian Journal of Mathematics

JF - Arabian Journal of Mathematics

IS - 4

ER -