Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-22T14:41:57.484Z Has data issue: false hasContentIssue false
  • English
  • Français

Transition phenomena for ladder epochs of random walks with small negative drift

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Vitali Wachtel
Vitali Wachtel*
Affiliation:
University of Munich
*
∗ Postal address: Mathematical Institute, University of Munich, Theresienstrasse 39, D-80333, Munich, Germany. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For a family of random walks {S(a)} satisfying E S1(a)=-a<0, we consider ladder epochs τ(a)=min {k≥1: Sk(a)<0}. We study the asymptotic behaviour, as a⇒0, of P (τ(a)>n) in the case when n=n(a)→∞. As a consequence, we also obtain the growth rates of the moments of τ(a).

Keywords

Random walkladder epochtransition phenomena

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60G52: Stable processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 4 , December 2009 , pp. 1189 - 1214
Copyright
Copyright © Applied Probability Trust 2009 

References

Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Bertoin, J. and Doney, R. A. (1996). Some asymptotic results for transient random walks. Adv. Appl. Prob. 28, 207–226.CrossRefGoogle Scholar
Doney, R. A. (1985). Conditional limit theorems for asymptotically stable random walks. Z. Wahrscheinlichkeitsth. 70, 351–360.Google Scholar
Doney, R. A. (1989). On the asymptotic behaviour of first passage times for transient random walk. Prob. Theory Relat. Fields 81, 239–246.Google Scholar
Embrechts, P. and Hawkes, J. (1982). A limit theorem for the tails of discrete infinitely divisible laws with application to fluctuation theory. J. Austral. Math. Soc. A 32, 412–422.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Gut, A. (1974). On the moments and limit distributions of some first passage times. Ann. Prob. 2, 277–308.Google Scholar
Keilson, J. (1963). The first passage time density for homogeneous skip-free walks on the continuum. Ann. Math. Statist. 34, 1003–1011.CrossRefGoogle Scholar
Lotov, V. I. (2006). On the mean value of the ladder epoch for random walks with small drift. Izv. Math. 70, 1225–1232.Google Scholar
Nagaev, A. V. (1969). Limit theorems for large deviations where Cramér's conditions are violated. Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk. 6, 17–22 (in Russian).Google Scholar
Nagaev, S. V. (1965). Some limit theorems for large deviations. Theory Prob. Appl. 10, 214–235.Google Scholar
Nagaev, S. V. (1979). Large deviations of sums of independent random variables. Ann. Prob. 7, 745–789.Google Scholar
Osipov, L. V. (1972). On probabilities of large deviations for sums of independent random variables. Theory Prob. Appl. 17, 309–331.Google Scholar
Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, New York.Google Scholar
Rogozin, B. A. (1971). On the distribution of the first ladder moment and height and fluctuations of a random walk. Theory Prob. Appl. 16, 575–595.Google Scholar
Rozovskiǐ, L. V. (1989). Probabilities of large deviations of sums of independent random variables with a common distribution function that belongs to the domain of attraction of the normal law. Theory Prob. Appl. 34, 625–644.Google Scholar
Spitzer, F. (2001). Principles of Random Walks. Springer, Berlin.Google Scholar
Zolotarev, V. M. (1957). Mellin–Stieltjes transform in probability theory. Theory Prob. Appl. 2, 433–460.Google Scholar