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 Tr. Mat. Inst. Steklova, 2013, Volume 282, Pages 22–41 (Mi tm3484)

Lower large deviations for supercritical branching processes in random environment

Vincent Bansayea, Christian Böinghoffb

a École Polytechnique, Palaiseau Cedex, France
b Fachbereich Mathematik, Goethe-Universität, Frankfurt am Main, Germany

Abstract: Branching processes in random environment $(Z_n\colon n\geq0)$ are the generalization of Galton–Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical regime, the process survives with a positive probability and grows exponentially on the non-extinction event. We focus on rare events when the process takes positive values but lower than expected. More precisely, we are interested in the lower large deviations of $Z$, which means the asymptotic behavior of the probability $\{1\leq Z_n\leq\exp(n\theta)\}$ as $n\to\infty$. We provide an expression for the rate of decrease of this probability under some moment assumptions, which yields the rate function. With this result we generalize the lower large deviation theorem of Bansaye and Berestycki (2009) by considering processes where $\mathbb P(Z_1=0\mid Z_0=1)>0$ and also much weaker moment assumptions.

DOI: https://doi.org/10.1134/S0371968513030035

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English version:
Proceedings of the Steklov Institute of Mathematics, 2013, 282, 15–34

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UDC: 519.218.27+519.214.8
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Citation: Vincent Bansaye, Christian Böinghoff, “Lower large deviations for supercritical branching processes in random environment”, Branching processes, random walks, and related problems, Collected papers. Dedicated to the memory of Boris Aleksandrovich Sevastyanov, corresponding member of the Russian Academy of Sciences, Tr. Mat. Inst. Steklova, 282, MAIK Nauka/Interperiodica, Moscow, 2013, 22–41; Proc. Steklov Inst. Math., 282 (2013), 15–34

Citation in format AMSBIB
\Bibitem{BanBoi13} \by Vincent~Bansaye, Christian~B\"oinghoff \paper Lower large deviations for supercritical branching processes in random environment \inbook Branching processes, random walks, and related problems \bookinfo Collected papers. Dedicated to the memory of Boris Aleksandrovich Sevastyanov, corresponding member of the Russian Academy of Sciences \serial Tr. Mat. Inst. Steklova \yr 2013 \vol 282 \pages 22--41 \publ MAIK Nauka/Interperiodica \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm3484} \crossref{https://doi.org/10.1134/S0371968513030035} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3308579} \elib{http://elibrary.ru/item.asp?id=20280543} \transl \jour Proc. Steklov Inst. Math. \yr 2013 \vol 282 \pages 15--34 \crossref{https://doi.org/10.1134/S0081543813060035} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000325961800003} \elib{http://elibrary.ru/item.asp?id=22212880} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84886072938} 

• http://mi.mathnet.ru/eng/tm3484
• https://doi.org/10.1134/S0371968513030035
• http://mi.mathnet.ru/eng/tm/v282/p22

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This publication is cited in the following articles:
1. N. Berestycki, N. Gantert, P. Moerters, N. Sidorova, “Galton-Watson trees with vanishing martingale limit”, J. Stat. Phys., 155:4 (2014), 737–762
2. Ch. Böinghoff, “Limit theorems for strongly and intermediately supercritical branching processes in random environment with linear fractional offspring distributions”, Stochastic Process. Appl., 124:11 (2014), 3553–3577
3. E. E. D'yakonova, “Limit theorem for multitype critical branching process evolving in random environment”, Discrete Math. Appl., 25:3 (2015), 137–147
4. Elena E. D'yakonova, “Reduced multitype critical branching processes in random environment”, Discrete Math. Appl., 28:1 (2018), 7–22
5. I. Grama, Q. Liu, E. Miqueu, “Berry–Esseen's bound and Cramér's large deviation expansion for a supercritical branching process in a random environment”, Stoch. Process. Their Appl., 127:4 (2017), 1255–1281
6. Ya. Wang, Q. Liu, “Limit theorems for a supercritical branching process with immigration in a random environment”, Sci. China-Math., 60:12 (2017), 2481–2502
7. I. Grama, Q. Liu, E. Miqueu, “Harmonic moments and large deviations for a supercritical branching process in a random environment”, Electron. J. Probab., 22 (2017), 99
8. N. Gantert, T. Hoefelsauer, “Large deviations for the maximum of a branching random walk”, Electron. Commun. Probab., 23 (2018), 34, 12 pp.
9. Damek E., Gantert N., Kolesko K., “Absolute Continuity of the Martingale Limit in Branching Processes in Random Environment”, Electron. Commun. Probab., 24 (2019), 42
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