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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2022, Volume 316, Pages 195–206
DOI: https://doi.org/10.4213/tm4245 (Mi tm4245) 		 
Conditional $L^1$-Convergence for the Martingale of a Critical Branching Process in Random Environment

Wenming Honga, Shengli Lianga, Xiaoyue Zhangb

a School of Mathematical Sciences & Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing, 100875, China
b School of Statistics, Capital University of Economics and Business, Beijing, 100070, China
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DOI: https://doi.org/10.4213/tm4245
Abstract: For a critical branching process $(Z_n)$ in a random environment $(\xi _n)$, a sufficient condition is given for the corresponding martingale ${Z_n}/{e^{S_n}}$ to converge in $L^1$ or to degenerate under $\mathbb P^+$, the probability under which the associated random walk is conditioned to stay nonnegative.
Keywords: branching process, random environment, multitype branching processes, change of measure, martingale convergence.
Funding agency Grant number
National Key Research and Development Program of China 2020YFA0712900
National Natural Science Foundation of China 11971062
12101419
Scientific Research Foundation for Young Teachers in the Capital University of Economics and Business XRZ2021035
This work was supported in part by the National Key Research and Development Program of China (no. 2020YFA0712900), NSFC (nos. 11971062, 12101419), and the Scientific Research Foundation for Young Teachers in the Capital University of Economics and Business (no. XRZ2021035).
Received: April 16, 2021
Revised: June 16, 2021
Accepted: November 3, 2021
English version:
Proceedings of the Steklov Institute of Mathematics, 2022, Volume 316, Pages 184–194
DOI: https://doi.org/10.1134/S0081543822010138
Bibliographic databases:
Document Type: Article
UDC: 519.218.2
Language: Russian
Citation: Wenming Hong, Shengli Liang, Xiaoyue Zhang, “Conditional $L^1$-Convergence for the Martingale of a Critical Branching Process in Random Environment”, Branching Processes and Related Topics, Collected papers. On the occasion of the 75th birthday of Andrei Mikhailovich Zubkov and 70th birthday of Vladimir Alekseevich Vatutin, Trudy Mat. Inst. Steklova, 316, Steklov Math. Inst., Moscow, 2022, 195–206; Proc. Steklov Inst. Math., 316 (2022), 184–194
Citation in format AMSBIB
\Bibitem{HonLiaZha22}
\by Wenming~Hong, Shengli~Liang, Xiaoyue~Zhang
\paper Conditional $L^1$-Convergence for the Martingale of a Critical Branching Process in Random Environment
\inbook Branching Processes and Related Topics
\bookinfo Collected papers. On the occasion of the 75th birthday of Andrei Mikhailovich Zubkov and 70th birthday of Vladimir Alekseevich Vatutin
\serial Trudy Mat. Inst. Steklova
\yr 2022
\vol 316
\pages 195--206
\publ Steklov Math. Inst.
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4245}
\crossref{https://doi.org/10.4213/tm4245}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2022
\vol 316
\pages 184--194
\crossref{https://doi.org/10.1134/S0081543822010138}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85129336378}
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    		Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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