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Limit theorems for critical branching processes in a finite-state-space Markovian environment

Published online by Cambridge University Press:  01 March 2022

Ion Grama*
Affiliation:
Université de Bretagne-Sud, CNRS UMR 6205, LMBA
Ronan Lauvergnat*
Affiliation:
Université de Bretagne-Sud, CNRS UMR 6205, LMBA
Émile Le Page*
Affiliation:
Université de Bretagne-Sud, CNRS UMR 6205, LMBA
*
*Postal address: Université de Bretagne-Sud, CNRS UMR 6205, Laboratoire de Mathématique de Bretagne Atlantique, Campus de Tohannic, BP573, 56017 Vannes, France.
*Postal address: Université de Bretagne-Sud, CNRS UMR 6205, Laboratoire de Mathématique de Bretagne Atlantique, Campus de Tohannic, BP573, 56017 Vannes, France.
*Postal address: Université de Bretagne-Sud, CNRS UMR 6205, Laboratoire de Mathématique de Bretagne Atlantique, Campus de Tohannic, BP573, 56017 Vannes, France.

Abstract

Let $(Z_n)_{n\geq 0}$ be a critical branching process in a random environment defined by a Markov chain $(X_n)_{n\geq 0}$ with values in a finite state space $\mathbb{X}$ . Let $ S_n = \sum_{k=1}^n \ln f_{X_k}^{\prime}(1)$ be the Markov walk associated to $(X_n)_{n\geq 0}$ , where $f_i$ is the offspring generating function when the environment is $i \in \mathbb{X}$ . Conditioned on the event $\{ Z_n>0\}$ , we show the nondegeneracy of the limit law of the normalized number of particles ${Z_n}/{e^{S_n}}$ and determine the limit of the law of $\frac{S_n}{\sqrt{n}} $ jointly with $X_n$ . Based on these results we establish a Yaglom-type theorem which specifies the limit of the joint law of $ \log Z_n$ and $X_n$ given $Z_n>0$ .

Type
Original Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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