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Conditional Limit Theorems for the Terms of a Random Walk Revisited

Published online by Cambridge University Press:  30 January 2018

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 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.

Type
Research Article
Copyright
© Applied Probability Trust 

Footnotes

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

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