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Conditional Limit Theorems for the Terms of a Random Walk Revisited

Part of: Limit theorems Special processes

Published online by Cambridge University Press:  30 January 2018

Shaul K. Bar-Lev ,
Ernst Schulte-Geers  and
Wolfgang Stadje
Shaul K. Bar-Lev*
Affiliation:
University of Haifa
Ernst Schulte-Geers*
Affiliation:
Federal Office for Information Security
Wolfgang Stadje*
Affiliation:
University of Osnabrück
*
∗ Postal address: Department of Statistics, University of Haifa, Haifa 31905, Israel. Email address: rbarlev@stat.haifa.ac.i
∗∗ Postal address: Federal Office for Information Security, Godesberger Allee 185-189, 53175 Bonn, Germany. Email address: ernst.schulte-geers@bsi.bund.de
∗∗∗ Postal address: Institute of Mathematics, University of Osnabrück, 49069 Osnabrück, Germany. Email address: wstadje@uos.de
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Abstract

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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 sn ≡ s is constant. We obtain convergence in total variation of PX1∣ Sn/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.

Keywords

Conditional limit theoremsums of i.i.d. random variablesrenewal theoryconvergence in total variationstable distribution

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60K05: Renewal theory
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 3 , September 2013 , pp. 871 - 882
Copyright
© Applied Probability Trust 

Footnotes

Supported by a Mercator professorship of the Deutsche Forschungsgemeinschaft at the University of Osnabrück.

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