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Conditional Limit Theorems for the Terms of a Random Walk Revisited

Part of: Special processes Limit theorems

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: [email protected]
∗∗ Postal address: Federal Office for Information Security, Godesberger Allee 185-189, 53175 Bonn, Germany. Email address: [email protected]
∗∗∗ Postal address: Institute of Mathematics, University of Osnabrück, 49069 Osnabrück, Germany. Email address: [email protected]
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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.

References

Asmussen, S. and Glynn, P. W. (2007). Stochastic Simulation. Algorithms and Analysis. Springer, New York.CrossRefGoogle Scholar
Diaconis, P. and Freedman, D. A. (1988). Conditional limit theorems for exponential families and finite versions of de Finetti's theorem. J. Theoret. Prob. 1, 381–410.Google Scholar
Doney, R. A. (1985). Conditional limit theorems for asymptotically stable random walks. Z. Wahrscheinlichkeitsth. 70, 351–360.Google Scholar
Doney, R. A. and O'Brien, G. L. (1992). Thickened renewal processes. Stoch. Process. Appl. 43, 1–8.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
Greenwood, L. and Perkins, E. (1983). A conditioned limit theorem for random walk and Brownian local time on square root boundaries. Ann. Prob. 11, 227–261.Google Scholar
Holst, L. (1979). Two conditional limit theorems with applications. Ann. Statist. 7, 551–557.Google Scholar
Holst, L. (1981). Some conditional limit theorems in exponential families. Ann. Prob. 9, 818–830.Google Scholar
Hooghiemstra, G. and Kennedy, D. P. (1984). Conditioned limit theorems and heavy traffic. Ann. Prob. 12, 631–634.CrossRefGoogle Scholar
Janson, S. (2001). Moment convergence in conditional limit theorems. J. Appl. Prob. 38, 421–437.Google Scholar
Martin-Löf, P. (1970). Statistical Models. Handwritten lecture notes for the academic year 1969–1970, compiled by Sundberg, R., Stockholm University.Google Scholar
Poleshchuk, O. M. (1995). A conditional limit theorem for a random walk with zero drift. Russian Math. Surveys 50, 446–448.CrossRefGoogle Scholar
Quine, M. P. (1985). Remainder term estimates in a conditional central limit theorem for integer-valued random variables. J. Austral. Math. Soc. Ser. A 39, 75–85.Google Scholar
Schulte-Geers, E. and Stadje, W. (1988). Some results on the Joint distribution of the renewal epochs prior to a given time instant. Stoch. Process. Appl. 30, 85–104.Google Scholar
Van Campenhout, J. M. and Cover, T. M. (1981). Maximum entropy and conditional probability. IEEE Trans. Inf. Theory 27, 483–489.Google Scholar
Yang, H. and Schrage, L. (2009). Conditions that cause risk pooling to increase inventory. Europ. J. Operat. Res. 192, 837–851.Google Scholar
Zabell, S. L. (1974). A limit theorem for conditional expectations with applications to probability theory and statistical mechanics. , Harvard University.Google Scholar
Zabell, S. L. (1993). A limit theorem for expectations conditional on a sum. J. Theoret. Prob. 6, 267–283.Google Scholar
Zolotarev, V. M. (1986). One-Dimensional Stable Distributions (Trans. Math. Monogr. 86). American Mathematical Society, Providence, RI.Google Scholar