The Annals of Applied Probability

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

Frèdèric Gosselin

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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

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

First available in Project Euclid: 27 August 2001

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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)

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


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.

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