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Article Dans Une Revue Advances in Applied Probability Année : 2021

LIMIT THEOREMS FOR CRITICAL BRANCHING PROCESSES IN A FINITE STATE SPACE MARKOVIAN ENVIRONMENT

Résumé

Let (Zn) n≥0 be a critical branching process in a random environment defined by a Markov chain (Xn) n≥0 with values in a finite state space X. Let Sn = n k=1 ln f X k (1) be the Markov walk associated to (Xn) n≥0 , where fi is the offspring generating function when the environment is i ∈ X. Conditioned on the event {Zn > 0}, we show the non degeneracy of the limit law of the normalized number of particles Zn/e Sn and determine the limit of the law of Sn √ n jointly with Xn. Based on these results we establish a Yaglom-type theorem which specifies the limit of the joint law of log Zn and Xn given Zn > 0.
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Dates et versions

hal-03435699 , version 1 (18-11-2021)

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  • HAL Id : hal-03435699 , version 1

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Ion Grama, Ronan Lauvergnat, Émile Le Page. LIMIT THEOREMS FOR CRITICAL BRANCHING PROCESSES IN A FINITE STATE SPACE MARKOVIAN ENVIRONMENT. Advances in Applied Probability, In press. ⟨hal-03435699⟩
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