Open Access
2008 Recurrence for branching Markov chains
Sebastian Müller
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Electron. Commun. Probab. 13: 576-605 (2008). DOI: 10.1214/ECP.v13-1424


The question of recurrence and transience of branching Markov chains is more subtle than for ordinary Markov chains; they can be classified in transience, weak recurrence, and strong recurrence. We review criteria for transience and weak recurrence and give several new conditions for weak recurrence and strong recurrence. These conditions make a unified treatment of known and new examples possible and provide enough information to distinguish between weak and strong recurrence. This represents a step towards a general classification of branching Markov chains. In particular, we show that in homogeneous cases weak recurrence and strong recurrence coincide. Furthermore, we discuss the generalization of positive and null recurrence to branching Markov chains and show that branching random walks on $Z$ are either transient or positive recurrent.


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Sebastian Müller. "Recurrence for branching Markov chains." Electron. Commun. Probab. 13 576 - 605, 2008.


Accepted: 24 November 2008; Published: 2008
First available in Project Euclid: 6 June 2016

zbMATH: 1187.60054
MathSciNet: MR2461533
Digital Object Identifier: 10.1214/ECP.v13-1424

Primary: 60J10
Secondary: 60J80

Keywords: branching Markov chains , positive recurrence , recurrence , spectral radius , strong recurrence , transience

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