Hostname: page-component-84b7d79bbc-rnpqb Total loading time: 0 Render date: 2024-07-27T16:36:05.923Z Has data issue: false hasContentIssue false
  • English
  • Français

Chernoff's theorem in the branching random walk

Published online by Cambridge University Press:  14 July 2016

J. D. Biggins
J. D. Biggins*
Affiliation:
University of Sheffield
Get access Rights & Permissions [Opens in a new window]

Abstract

If Fn∗ is the n-fold Stieltjes convolution of the increasing function F, then a version of Chernoff's theorem, on the limiting behaviour of (Fn∗(na))1/n, is established for Fn∗. If Z(n)(t) is the number of the nth-generation people to the left of t in a supercritical branching random walk then an analogous result is proved for Z(n).

Keywords

CHERNOFF'S THEOREMBRANCHING RANDOM WALKBRANCHING PROCESSES
Type
Short Communications
Information
Journal of Applied Probability , Volume 14 , Issue 3 , September 1977 , pp. 630 - 636
Copyright
Copyright © Applied Probability Trust 1977 

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

Bahadur, R. R. and Rao, R. Ranga (1960) On deviation of the sample mean. Ann. Math. Statist. 31, 1015–1027.CrossRefGoogle Scholar
Biggins, J. D. (1976) The first and last-birth problems for a multitype age-dependent branching process. Adv. Appl. Prob. 8, 446–459.Google Scholar
Chernoff, H. (1952) A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statist. 23, 493–507.Google Scholar
Hammersley, J. M. (1974) Postulates for subadditive processes. Ann. Prob. 2, 652–680.Google Scholar
Harris, T. E. (1963) The Theory of Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
Jagers, P. (1975) Branching Processes with Biological Applications. Wiley, London.Google Scholar
Kingman, J. F. C. (1975) The first birth problem for an age-dependent branching process. Ann. Prob. 12, 341–345.Google Scholar
Rockafellar, R. T. (1970) Convexity Analysis. Princeton University Press, Princeton, N.J.CrossRefGoogle Scholar