On series of discrete random variables, 1: Real trinomial distributions with fixed probabilities

Jean Marc Deshouillers, Gregory A. Freiman, William Moran

Research output: Contribution to journalArticlepeer-review

Abstract

This paper begins the study of the local limit behaviour of triangular arrays of independent random variables (ζn,k)1≤k≤n where the law of ζn,k depends on on n. We consider the case when ζn,1 takes three integral values 0 < a1(n) < a2(n) with respective probabilities p0,p1,p2 which do not depend on n. We show three types of limit behaviours for the sequence of r. v. ηn = ζn,1 + ⋯ + ζn,n, according as a2(n)/gcd(a1(n), a2(n)) tends to infinity slower, quicker or at the same speed as √n.

Original languageEnglish
Pages (from-to)411-423
Number of pages13
JournalAsterisque
Volume258
StatePublished - 1999

Keywords

  • Characteristic function
  • Circle method
  • Local limit theorems
  • Number theoretic methods
  • Sums of discrete random variables

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