Conditional limit theorems for the terms of a randomwalk revisited

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

doi:10.1239/jap/1378401242

Conditional limit theorems for the terms of a randomwalk revisited

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

Department of Statistics

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper we derive limit theorems for the conditional distribution of X₁ given S_n = s_n as n → ∞, where the X_iare independent and identically distributed (i.i.d.) random variables, S_n = X₁ + ... + X_n, and s _n/n converges or s_n ≡ s is constant. We obtain convergence in total variation of ℙ_nX₁ | S _n/n=s to a distribution associated to that of X₁ and of ℙ_nX₁ | S_n=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 language	English
Pages (from-to)	871-882
Number of pages	12
Journal	Journal of Applied Probability
Volume	50
Issue number	3
DOIs	https://doi.org/10.1239/jap/1378401242
State	Published - Sep 2013

Keywords

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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10.1239/jap/1378401242

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abstract = "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.",

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AB - 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.

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KW - Convergence in total variation

KW - Renewal theory

KW - Stable distribution

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Conditional limit theorems for the terms of a randomwalk revisited

Abstract

Keywords

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