Journal of Applied Probability

Conditional limit theorems for the terms of a random walk revisited

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

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In this paper we derive limit theorems for the conditional distribution of X1 given Sn=sn as n→ ∞, where the Xi are independent and identically distributed (i.i.d.) random variables, Sn=X1+··· +Xn, and sn/n converges or sns is constant. We obtain convergence in total variation of PX1Sn/n=s to a distribution associated to that of X1 and of PnX1Sn=s to a gamma distribution. The case of stable distributions (to which the method of associated distributions cannot be applied) is studied in detail.

Article information

J. Appl. Probab., Volume 50, Number 3 (2013), 871-882.

First available in Project Euclid: 5 September 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60K05: Renewal theory

Conditional limit theorem sums of i.i.d. random variables renewal theory convergence in total variation stable distribution


Bar-Lev, Shaul K.; Schulte-Geers, Ernst; Stadje, Wolfgang. Conditional limit theorems for the terms of a random walk revisited. J. Appl. Probab. 50 (2013), no. 3, 871--882. doi:10.1239/jap/1378401242.

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