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

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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 languageEnglish
JournalJournal of Theoretical Probability
DOIs
Publication statusPublished - 2019 Jan 1

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P-adic
Large Deviations
Identically distributed
Random variable
Tail Probability
Rate Function
Scaling
Sums of Random Variables
Stable Laws
Large Deviation Principle
Accumulate
Logarithm
Infinity
Converge
Large deviations
Tail probability
Random variables
Sums of random variables

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

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title = "Large Deviations for Scaled Sums of p-Adic-Valued Rotation-Symmetric Independent and Identically Distributed Random Variables",
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.",
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language = "English",
journal = "Journal of Theoretical Probability",
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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.

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