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Some asymptotic results for transient random walks

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

J. Bertoin  and
R. A. Doney
J. Bertoin*
Affiliation:
Université Paris VI
R. A. Doney*
Affiliation:
University of Manchester
*
* Postal address: Laboratoire de Probabilités (CNRS), Université Paris VI, 4 Place Jussieu, 75252 Paris Cedex 05, France.
** Postal address: Statistical Laboratory, Department of Mathematics, University of Manchester, M13 9PL, UK.
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Abstract

We consider a real-valued random walk S which drifts to –∞ and is such that E(exp θS1) < ∞ for some θ > 0, but for which Cramér's condition fails. We investigate the asymptotic tail behaviour of the distributions of the all time maximum, the upwards and downwards first passage times and the last passage times. As an application, we obtain new limit theorems for certain conditional laws.

Keywords

TRANSIENT RANDOM WALKS

MSC classification

Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 28 , Issue 1 , March 1996 , pp. 207 - 226
Copyright
Copyright © Applied Probability Trust 1996 

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References

[1] Bertoin, J. and Doney, R. A. (1994). On conditioning a random walk to stay non-negative. Ann. Prob. 22, 2152–2167.CrossRefGoogle Scholar
[2] Bertoin, J. and Doney, R. A. (1994) On the local behaviour of ladder height distributions. J. Appl. Prob. 31, 816–821.CrossRefGoogle Scholar
[3] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987) Regular Variation. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
[4] Borovkov, A. A. (1976) Stochastic Processes in Queuing Theory. Springer, Berlin.CrossRefGoogle Scholar
[5] Brown, L. D. (1986) Fundamentals of Statistical Exponential Families, with Applications in Statistical Decision Theory vol. 9. Institute of Mathematical Statistics, Hayward.CrossRefGoogle Scholar
[6] Chover, J., Ney, P. and Wainger, S. (1973) Functions of probability measures. J. Anal. Math. 26, 255–302.CrossRefGoogle Scholar
[7] Chover, J., Ney, P. and Wainger, S. (1973) Degeneracy properties of subcritical branching processes. Ann. Prob. 1, 663–673.CrossRefGoogle Scholar
[8] Doney, R. A. (1989) On the asymptotic behaviour of first passage times for transient random walks. Z. Wahrscheinlischkeitsth. 81, 239–246.Google Scholar
[9] Doney, R. A. (1989) Last-exit times for random walks. Stoch. Proc. Appl. 31, 321–331.CrossRefGoogle Scholar
[10] Doney, R. A. (1989) A large deviation local limit theorem. Math. Proc. Cambridge Phil. Soc. 105, 575–577.CrossRefGoogle Scholar
[11] Embrechts, P. and Goldie, C. M. (1982) On convolution tails. Stoch. Proc. Appl. 13, 263–278.CrossRefGoogle Scholar
[12] Embrechts, P. and Goldie, C. M. (1980) On closure and factorization properties of subexponential and related distributions. J. Australian Math. Soc. A 29, 243–256.CrossRefGoogle Scholar
[13] Embrechts, P. and Hawkes, J. (1982) A limit theorem for the tails of discrete infinitely divisible distributions with applications to fluctuation theory. J. Australian Math. Soc. A 32, 412–422.CrossRefGoogle Scholar
[14] Embrechts, P. and Veraverbeke, N. (1982) Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance: Math. Econ. 1, 55–72.Google Scholar
[15] Feller, W. E. (1972) An Introduction to Probability Theory and its Applications. Vol. II, 2nd edn. Wiley, New York.Google Scholar
[16] Grübel, R. (1995) Tail behaviour of ladder height distributions in random walks. J. Appl. Prob. 22, 705–709.CrossRefGoogle Scholar
[17] Grübel, R. (1987) On subordinated distributions and generalized renewal measures. Ann. Prob. 15, 394–415.CrossRefGoogle Scholar
[18] Iglehart, D. L. (1972) Extreme values for the GI/G/1 queue. Ann. Math. Statist. 43, 627–635.CrossRefGoogle Scholar
[19] Keener, R. W. (1992) Limit theorems for random walks conditioned to stay positive. Ann. Prob. 20, 801–824.CrossRefGoogle Scholar
[20] Nagaev, S. V. (1982) On the asymptotic behaviour of one-sided large deviation probabilities. Theory Prob. Appl. 26, 362–366.CrossRefGoogle Scholar
[21] Petrov, V. V. (1965) On the probabilities of large deviations for sums of independent random variables. Theory Prob. Appl. 10, 287–297.CrossRefGoogle Scholar
[22] Teugels, J. L. (1975) The class of subexponential distributions. Ann. Prob. 3, 1000–1011.CrossRefGoogle Scholar
[23] Veraverbeke, N. (1977) Asymptotic behaviour of Wiener-Hopf factors of a random walk. Stoch. Proc. Appl. 5, 27–37.CrossRefGoogle Scholar
[24] Veraverbeke, N. and Teugels, J. L. (1975) The exponential rate of convergence of the maximum of a random walk. J. Appl. Prob. 12, 279–288.CrossRefGoogle Scholar