Abstract
We consider multitype branching processes evolving in a Markovian random environment. To determine whether or not the branching process becomes extinct almost surely is akin to computing the maximal Lyapunov exponent of a sequence of random matrices, which is a notoriously difficult problem. We define Markov chains associated to the branching process, and we construct bounds for the Lyapunov exponent. The bounds are obtained by adding or by removing information: to add information results in a lower bound, to remove information results in an upper bound, and we show that adding less information improves the lower bound. We give a few illustrative examples and we observe that the upper bound is generally more accurate than the lower bounds.
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Acknowledgments
Our paper has benefited from interesting questions and comments from two anonymous referees. Sophie Hautphenne is supported by the Australian Research Council (ARC), Grants FL130100039 and DE150101044. Guy Latouche acknowledges the support of the Ministère de la Communauté française de Belgique through the ARC Grant AUWB-08/13–ULB 5.
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Hautphenne, S., Latouche, G. Lyapunov Exponents for Branching Processes in a Random Environment: The Effect of Information. J Stat Phys 163, 393–410 (2016). https://doi.org/10.1007/s10955-016-1474-3
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DOI: https://doi.org/10.1007/s10955-016-1474-3
Keywords
- Product of random matrices
- Lyapunov exponent
- Multitype branching process
- Markovian random environment
- Extinction criterion