Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-24T01:26:54.496Z Has data issue: false hasContentIssue false
  • English
  • Français

Path to survival for the critical branching processes in a random environment

Part of: Stochastic processes Markov processes Special processes Limit theorems

Published online by Cambridge University Press:  22 June 2017

Vladimir Vatutin  and
Elena Dyakonova
Vladimir Vatutin*
Affiliation:
Steklov Mathematical Institute
Elena Dyakonova*
Affiliation:
Steklov Mathematical Institute
*
* Postal address: Steklov Mathematical Institute, Gubkin Street 8, Moscow 119991, Russia.
* Postal address: Steklov Mathematical Institute, Gubkin Street 8, Moscow 119991, Russia.
Get access Rights & Permissions [Opens in a new window]

Abstract

A critical branching process {Zk, k = 0, 1, 2, ...} in a random environment is considered. A conditional functional limit theorem for the properly scaled process {log Zpu, 0 ≤ u < ∞} is established under the assumptions that Zn > 0 and p ≪ n. It is shown that the limiting process is a Lévy process conditioned to stay nonnegative. The proof of this result is based on a limit theorem describing the distribution of the initial part of the trajectories of a driftless random walk conditioned to stay nonnegative.

Keywords

Branching processrandom environmentrandom walk conditioned to stay positiveLévy process conditioned to stay positivechange of measurefunctional limit theorem

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60K37: Processes in random environments 60G50: Sums of independent random variables; random walks 60F17: Functional limit theorems; invariance principles
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 2 , June 2017 , pp. 588 - 602
Copyright
Copyright © Applied Probability Trust 2017 

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

[1] Afanasyev, V. I. (1993). A limit theorem for a critical branching process in random environment. Diskret. Mat. 5, 45–58 (in Russian). Google Scholar
[2] Afanasyev, V. I. (1997). A new theorem for a critical branching process in random environment. Discrete Math. Appl. 7, 497–513. Google Scholar
[3] Afanasyev, V. I. (1999). On the maximum of a critical branching process in a random environment. Discrete Math. Appl. 9, 267–284. Google Scholar
[4] Afanasyev, V. I. (1999). On the time of reaching a fixed level by a critical branching process in a random environment. Discrete Math. Appl. 9, 627–643. Google Scholar
[5] Afanasyev, V. I. (2001). A functional limit theorem for a critical branching process in a random environment. Discrete Math. Appl. 11, 587–606. Google Scholar
[6] Afanasyev, V. I., Böinghoff, C., Kersting, G. and Vatutin, V. A. (2012). Limit theorems for weakly subcritical branching processes in random environment. J. Theoret. Prob. 25, 703–732. Google Scholar
[7] Afanasyev, V. I., Böinghoff, C., Kersting, G. and Vatutin, V. A. (2014). Conditional limit theorems for intermediately subcritical branching processes in random environment. Ann. Inst. H. Poincaré Prob. Statist. 50, 602–627. Google Scholar
[8] Afanasyev, V. I., Geiger, J., Kersting, G. and Vatutin, V. A. (2005). Criticality for branching processes in random environment. Ann. Prob. 33, 645–673. Google Scholar
[9] Afanasyev, V. I., Geiger, J., Kersting, G. and Vatutin, V. A. (2005). Functional limit theorems for strongly subcritical branching processes in random environment. Stoch. Process Appl. 115, 1658–1676. Google Scholar
[10] Bertoin, J. and Doney, R. A. (1994). On conditioning a random walk to stay nonnegative. Ann. Prob. 22, 2152–2167. Google Scholar
[11] Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Willey, New York. CrossRefGoogle Scholar
[12] Böinghoff, C., Dyakonova, E. E., Kersting, G. and Vatutin, V. A. (2010). Branching processes in random environment which extinct at a given moment. Markov Process. Relat. Fields 16, 329–350. Google Scholar
[13] Caravenna, F. and Chaumont, L. (2008). Invariance principles for random walks conditioned to stay positive. Ann. Inst. H. Poincaré Prob. Statist. 44, 170–190. Google Scholar
[14] Chaumont, L. (1996). Conditionings and path decompositions for Lévy processes. Stoch. Process. Appl. 64, 39–54. Google Scholar
[15] Chaumont, L. (1997). Excursion normalisée, méandre at pont pour les processus de Lévy stables. Bull. Sci. Math. 121, 377–403. Google Scholar
[16] Doney, R. A. (2012). Local behavior of first passage probabilities. Prob. Theory Relat. Fields 152, 559–588. Google Scholar
[17] Dyakonova, E. E., Geiger, J. and Vatutin, V. A. (2004). On the survival probability and a functional limit theorem for branching processes in random environment. Markov Process. Relat. Fields 10, 289–306. Google Scholar
[18] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York. Google Scholar
[19] Geiger, J. and Kersting, G. (2002). The survival probability of a critical branching process in random environment. Theory Prob. Appl. 45, 518–526. Google Scholar
[20] Kozlov, M. V. (1976). On the asymptotic behavior of the probability of non-extinction for critical branching processes in a random environment. Theory Prob. Appl. 21, 791–804. Google Scholar
[21] Kozlov, M. V. (1995). A conditional function limit theorem for a critical branching process in a random environment. Dokl. Akad. Nauk 344, 12–15 (in Russian). Google Scholar
[22] Seneta, E. (1976). Regularly Varying Functions (Lecture Notes Math. 508). Springer, Berlin. Google Scholar
[23] Vatutin, V. A. (2003). Reduced branching processes in a random environment: the critical case. Theory Prob. Appl. 47, 99–113. Google Scholar
[24] Vatutin, V. A. (2016). Subcritical branching processes in random environment. In Branching Processes and Their Applications (Lecture Notes Statist. 219), eds I. M. del Puerto et al., Springer, Cham, pp. 97–115. Google Scholar
[25] Vatutin, V. A. and Dyakonova, E. E. (2004). Galton–Watson branching processes in a random environment. I. Limit theorems. Theory Prob. Appl. 48, 314–336. Google Scholar
[26] Vatutin, V. and Liu, Q. (2015). Limit theorems for decomposable branching processes in a random environment. J. Appl. Prob. 52, 877–. Google Scholar
[27] Vatutin, V. A. and Wachtel, V. (2009). Local probabilities for random walks conditioned to stay positive. Prob. Theory Relat. Fields 143, 177–217. Google Scholar
[28] Vatutin, V. A., Dyakonova, E. E. and Sagitov, S. (2013). Evolution of branching processes in a random environment. Proc. Steklov Inst. Math. 282, 220–242. Google Scholar
[29] Zolotarev, V. M. (1957). Mellin–Stieltjes transform in probability theory. Theory Prob. Appl. 2, 433–460. Google Scholar