Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T19:21:40.725Z Has data issue: false hasContentIssue false
  • English
  • Français

Reduced subcritical Galton-Watson processes in a random environment

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Klaus Fleischmann  and
Vladimir A. Vatutin
Klaus Fleischmann*
Affiliation:
Weierstrass Institute, Berlin
Vladimir A. Vatutin*
Affiliation:
Steklov Mathematical Institute, Moscow
*
∗ Postal address: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D-10117 Berlin, Germany. Email address: fleischmann@wias-berlin.de
∗∗ Postal address: Department of Discrete Mathematics, Steklov Mathematical Institute, 8 Gubkin Street, 117 966 Moscow, GSP-1, Russia.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the structure of genealogical trees of reduced subcritical Galton-Watson processes in a random environment assuming that all (randomly varying in time) offspring generating functions are fractional linear. We show that this structure may differ significantly from that of the ‘classical’ reduced subcritical Galton-Watson processes. In particular, it may look like a complex ‘hybrid’ of classical reduced super and subcritical processes. Some relations with random walks in a random environment are discussed.

Keywords

Random environmentbranching processrandom walkreduced processreduced treeconditional limit theoremsource timehybrid behaviorladder epochs

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 1 , March 1999 , pp. 88 - 111
Copyright
Copyright © Applied Probability Trust 1999 

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.)

Footnotes

Supported in part by the Grant RFBR N 96-00338, INTAS-RFBR 95-0099, and the DFG.

References

Afanasyev, V. I. (1980). Limit theorems for a conditional random walk and some applications. Diss. Cand. Sci., Moscow, MSU.Google Scholar
Afanasev, V. I. (1990). On a maximum of a transient random walk in random environment. Theory Prob. Appl. 35, 205–215.CrossRefGoogle Scholar
Afanasyev, V. I. (1998). Limit theorems for a moderately subcritical branching process in a random environment. Discr. Math. Appl. 8, 35–52.Google Scholar
Agresti, A. (1975). On the extinction time of varying and random environment branching processes. J. Appl. Prob. 12, 39–46.Google Scholar
Athreya, K. B. and Karlin, S. (1972). On branching processes in random environment: II. Limit theorems. Ann. Math. Statist. 42, 1843–1858.Google Scholar
Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
Borovkov, K. A. and Vatutin, V. A. (1997). Reduced critical branching processes in random environment. Stoch. Proc. Appl. 71, 225–240.CrossRefGoogle Scholar
Durrett, R. (1978). The genealogy of critical branching processes. Stoch. Proc. Appl. 8, 101–116.Google Scholar
Dwass, M. (1969). The total progeny in a branching process and related random walk. J. Appl. Prob. 6, 682–686.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and its Applications. Vol. II, 2nd edn. Wiley, New York.Google Scholar
Fleischmann, K. and Prehn, U. (1974). Ein Grenzwertsatz für subkritische Verzweigungsprozesse mit endlich vielen Typen von Teilchen. Math. Nachr. 64, 357–362.CrossRefGoogle Scholar
Fleischmann, K. and Siegmund-Schultze, R. (1977). The structure of reduced critical Galton-Watson processes. Math. Nachr. 79, 233–241.Google Scholar
Jagers, P., Nerman, O. and Taib, Z. (1991). When did Joe's great …grandfather live? Or on the timescale of evolution. In Selected Proceedings of the Sheffield Symposium on Applied Probability, eds. Basawa, I. V. and Taylor, R. L. (IMS Lecture Notes. Monogr. Ser. 18). Inst. Math. Statist., Hayward, CA.Google Scholar
Kao, P. (1978). Limiting diffusion for random walks with drift conditioned to stay positive. J. Appl. Prob. 15, 280–291.Google Scholar
Kesten, H., Kozlov, M. V. and Spitzer, F. (1975). A limit theorem for random walk in a random environment. Composito Mathematica 30, 145–168.Google Scholar
Kozlov, M. V. (1973). Random walk in a one-dimensional random environment. Theory Prob. Appl. 18, 387–389.CrossRefGoogle Scholar
Kozlov, M. V. (1976). On the asymptotic probability of non-extinction for a critical branching process in random environment. Theory Prob. Appl. 21, 791–804.Google Scholar
O'Connell, N. (1995). The genealogy of branching processes and the age of our most recent common ancestor. Adv. Appl. Prob. 27, 418–442.Google Scholar
Prehn, U. (1979). Die mittlere Quellzeit subkritischer Verzweigungsprozesse. Math. Nachr. 88, 409–410.Google Scholar
Sagitov, S. (1995). Three limit theorems for reduced critical branching processes. Russian Math. Surv. 50, 1025–1043.Google Scholar
Sinai, J. G. (1982). Limit behavior of one-dimensional random walks in a random medium. Theory Prob. Appl. 27, 256–268.Google Scholar
Smith, W. L. and Wilkinson, W. E. (1969). On branching processes in random environments. Ann. Math. Statist. 40, 814–827.CrossRefGoogle Scholar
Vatutin, V. A. (1979). Distance to the nearest common ancestor in Bellman-Harris branching processes. Math. Notes 25, 378–387.CrossRefGoogle Scholar
Vatutin, V. A. and D'yakonova, E. E. (1998). On the extinction probability at a given moment for a class of branching processes in a random environment (in Russian). Diskretnaya Mathematika 9, 100–126.Google Scholar
Veraverbeke, N. and Teugels, J. L. (1976). The exponential rate of convergence of the distribution of the maximum of a random walk. J. Appl. Prob. 13, 733–740.Google Scholar
Vigilant, L., Pennington, R., Harpending, H., Kocher, T. D. and Wilson, A. (1989). Mitochondrial DNA sequences in single hairs from a southern African population. Proc. Nat. Acad. Sci. USA 86, 9350–9354.Google Scholar
Zubkov, A. M. (1975). Limiting distributions for the distance to the closest mutual ancestor. Theory Prob. Appl. 20, 602–612.Google Scholar