Journal of Statistical Physics

, Volume 163, Issue 2, pp 393–410 | Cite as

Lyapunov Exponents for Branching Processes in a Random Environment: The Effect of Information

Article

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.

Keywords

Product of random matrices Lyapunov exponent Multitype branching process Markovian random environment Extinction criterion 

Mathematics Subject Classification

60J80 60J22 60J27 

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsThe University of MelbourneMelbourneAustralia
  2. 2.Institute of MathematicsEcole Polytechnique Fédérale de LausanneLausanneSwitzerland
  3. 3.Département d’informatiqueUniversité libre de BruxellesBrusselsBelgium

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