Conditional limit theorems for the terms of a randomwalk revisited

Shaul K. Bar-Lev, Ernst Schulte-Geers, Wolfgang Stadje

Research output: Contribution to journalArticlepeer-review


In this paper we derive limit theorems for the conditional distribution of X1 given Sn = sn as n → ∞, where the Xiare independent and identically distributed (i.i.d.) random variables, Sn = X1 + ... + Xn, and s n/n converges or sn ≡ s is constant. We obtain convergence in total variation of ℙnX1 | S n/n=s to a distribution associated to that of X1 and of ℙnX1 | Sn=s to a gamma distribution. The case of stable distributions (to which the method of associated distributions cannot be applied) is studied in detail.

Original languageEnglish
Pages (from-to)871-882
Number of pages12
JournalJournal of Applied Probability
Issue number3
StatePublished - Sep 2013


  • Conditional limit theorem
  • Convergence in total variation
  • Renewal theory
  • Stable distribution
  • Sums of i.i.d. random variables

ASJC Scopus subject areas

  • Statistics and Probability
  • General Mathematics
  • Statistics, Probability and Uncertainty


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