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Yaglom limits can depend on the starting state

  • R. D. Foley (a1) and D. R. McDonald (a2)

Abstract

We construct a simple example, surely known to Harry Kesten, of an R-transient Markov chain on a countable state space S ∪ {δ}, where δ is absorbing. The transition matrix K on S is irreducible and strictly substochastic. We determine the Yaglom limit, that is, the limiting conditional behavior given nonabsorption. Each starting state xS results in a different Yaglom limit. Each Yaglom limit is an R -1-invariant quasi-stationary distribution, where R is the convergence parameter of K. Yaglom limits that depend on the starting state are related to a nontrivial R -1-Martin boundary.

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Corresponding author

* Postal address: Department of Industrial & Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0205, USA. Email address: rfoley@gatech.edu
** Postal address: Department of Mathematics and Statistics, The University of Ottawa, Ottawa, Ontario, K1N 6N5, Canada. Email address: dmdsg@uottawa.ac

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