Electronic Communications in Probability

Recurrence for branching Markov chains

Sebastian Müller

Full-text: Open access

Abstract

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.

Article information

Source
Electron. Commun. Probab., Volume 13 (2008), paper no. 54, 576-605.

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465233481

Digital Object Identifier
doi:10.1214/ECP.v13-1424

Mathematical Reviews number (MathSciNet)
MR2461533

Zentralblatt MATH identifier
1187.60054

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

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

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Müller, Sebastian. Recurrence for branching Markov chains. Electron. Commun. Probab. 13 (2008), paper no. 54, 576--605. doi:10.1214/ECP.v13-1424. https://projecteuclid.org/euclid.ecp/1465233481


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