Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-05T23:24:14.722Z Has data issue: false hasContentIssue false
  • English
  • Français

The maximum and time to absorption of a left-continuous random walk

Published online by Cambridge University Press:  14 July 2016

P. J. Green
P. J. Green*
Affiliation:
University of Bath
Get access Rights & Permissions [Opens in a new window]

Abstract

For a left-continuous random walk, absorbing at 0, the joint distribution of the maximum and time to absorption is derived. A description of the tails of the distributions and a conditional limit theorem are obtained for the cases where absorption is certain.

Keywords

BRANCHING PROCESSMAXIMUMTOTAL PROGENYLIMIT THEOREMSLEFT-CONTINUOUS RANDOM WALK
Type
Research Papers
Information
Journal of Applied Probability , Volume 13 , Issue 3 , September 1976 , pp. 444 - 454
Copyright
Copyright © Applied Probability Trust 1976 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ahlfors, L. V. (1966) Complex Analysis, 2nd edn. McGraw-Hill, New York.Google Scholar
Dwass, M. (1969) The total progeny in a branching process and a related random walk. J. Appl. Prob. 6, 682–686.Google Scholar
Harris, T. E. (1963) The Theory of Branching Processes. Springer, Berlin.Google Scholar
Kozakiewicz, W. (1947) On the convergence of sequences of moment generating functions. Ann. Math. Statist. 18, 61–69.Google Scholar
Lindvall, T. (1975) On the maximum of a branching process. Technical Report, Chalmers University of Technology, Göteborg, Sweden.Google Scholar
Pakes, A. G. (1973) Conditional limit theorems for a left-continuous random walk. J. Appl. Prob. 10, 39–53.Google Scholar
Pakes, A. G. (1974) Some limit theorems for Markov chains with applications to branching processes. In Studies in Probability and Statistics: Papers in Honour of E. J. G. Pitman, ed. Williams, E. J. Jerusalem Academic Press, Jerusalem.Google Scholar
Seneta, E. (1967) On the maxima of absorbing Markov chains. Austral. J. Statist. 9, 93–102.Google Scholar
Spitzer, F. (1964) Principles of Random Walk. Van Nostrand, Princeton.Google Scholar
Takács, L. (1967) Combinatorial Methods in the Theory of Stochastic Processes. Wiley, New York.Google Scholar
Zolotarev, V. M. (1954) On a problem from the theory of branching stochastic processes (in Russian). Uspekhi Math. Nauk. 9, Number 2, 147–156.Google Scholar