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

Part of: Special processes Limit theorems Markov processes Stochastic processes

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.
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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 

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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