The Annals of Applied Probability

Aysmptotic Behavior of Absorbing Markov Chains Conditional on Nonabsorption for Applications in Conservation Biology

Frèdèric Gosselin

Full-text: Open access

Abstract

We find a Lyapunov-type sufficient condition for discrete-time Markov chains on a countable state space including an absorbing set to almost surely reach this absorbing set and to asymptotically stabilize conditional on nonabsorption. This result is applied to Bienaymè-Galton-Watson-like branching processes in which the offspring distribution depends on the current population size. This yields a generalization of the Yaglom limit. The techniques used mainly rely on the spectral theory of linear operators on Banach spaces and especially on the notion of quasi-compact linear operator.

Article information

Source
Ann. Appl. Probab., Volume 11, Number 1 (2001), 261-284.

Dates
First available in Project Euclid: 27 August 2001

Permanent link to this document
https://projecteuclid.org/euclid.aoap/998926993

Digital Object Identifier
doi:10.1214/aoap/998926993

Mathematical Reviews number (MathSciNet)
MR1825466

Zentralblatt MATH identifier
1019.60082

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47B65: Positive operators and order-bounded operators 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40] 60J85: Applications of branching processes [See also 92Dxx] 92D25: Population dynamics (general)

Keywords
homogeneous Markov chain extinction absorbing set quasi-stationary distribution Yaglom limit density-dependence population-size-dependent Bienaymè-Galton-Watson branching processes quasi-compact linear operator nonnegative operator irreducible matrix infinite dimensional matrix spectral theory conservation biology population viability analysis demography population dynamics Lyapunov-type condition

Citation

Gosselin, Frèdèric. Aysmptotic Behavior of Absorbing Markov Chains Conditional on Nonabsorption for Applications in Conservation Biology. Ann. Appl. Probab. 11 (2001), no. 1, 261--284. doi:10.1214/aoap/998926993. https://projecteuclid.org/euclid.aoap/998926993


Export citation