### Abstract

The law of an appropriately scaled sum of p-adic-valued, independent, identically and rotation-symmetrically distributed random variables weakly converges to a semi-stable law, if the tail probabilities of the variables satisfy some assumption. If we consider a scaled sum of such random variables with a sufficiently much higher scaling order, it accumulates to the origin, and the mass of any set not including the origin gets small. The purpose of this article is to investigate the asymptotic order of the logarithm of the mass of such sets off the origin. The order is explicitly given under some assumptions on the tail probabilities of the random variables and the scaling order of their sum. It is also proved that the large deviation principle follows with a rate function being constant except at the origin, and the rate function is good only for the case its value is infinity off the origin.

Original language | English |
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Journal | Journal of Theoretical Probability |

DOIs | |

Publication status | Published - 2019 Jan 1 |

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### Keywords

- Large deviations
- Limit theorem
- p-adic field
- Scaled sum of independent and identically distributed

### ASJC Scopus subject areas

- Statistics and Probability
- Mathematics(all)
- Statistics, Probability and Uncertainty

### Cite this

**Large Deviations for Scaled Sums of p-Adic-Valued Rotation-Symmetric Independent and Identically Distributed Random Variables.** / Yasuda, Kumi.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Large Deviations for Scaled Sums of p-Adic-Valued Rotation-Symmetric Independent and Identically Distributed Random Variables

AU - Yasuda, Kumi

PY - 2019/1/1

Y1 - 2019/1/1

N2 - The law of an appropriately scaled sum of p-adic-valued, independent, identically and rotation-symmetrically distributed random variables weakly converges to a semi-stable law, if the tail probabilities of the variables satisfy some assumption. If we consider a scaled sum of such random variables with a sufficiently much higher scaling order, it accumulates to the origin, and the mass of any set not including the origin gets small. The purpose of this article is to investigate the asymptotic order of the logarithm of the mass of such sets off the origin. The order is explicitly given under some assumptions on the tail probabilities of the random variables and the scaling order of their sum. It is also proved that the large deviation principle follows with a rate function being constant except at the origin, and the rate function is good only for the case its value is infinity off the origin.

AB - The law of an appropriately scaled sum of p-adic-valued, independent, identically and rotation-symmetrically distributed random variables weakly converges to a semi-stable law, if the tail probabilities of the variables satisfy some assumption. If we consider a scaled sum of such random variables with a sufficiently much higher scaling order, it accumulates to the origin, and the mass of any set not including the origin gets small. The purpose of this article is to investigate the asymptotic order of the logarithm of the mass of such sets off the origin. The order is explicitly given under some assumptions on the tail probabilities of the random variables and the scaling order of their sum. It is also proved that the large deviation principle follows with a rate function being constant except at the origin, and the rate function is good only for the case its value is infinity off the origin.

KW - Large deviations

KW - Limit theorem

KW - p-adic field

KW - Scaled sum of independent and identically distributed

UR - http://www.scopus.com/inward/record.url?scp=85064216634&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85064216634&partnerID=8YFLogxK

U2 - 10.1007/s10959-019-00894-0

DO - 10.1007/s10959-019-00894-0

M3 - Article

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

SN - 0894-9840

ER -