Yaglom limits can depend on the starting state

Abstract

AbstractWe construct a simple example, surely known to Harry Kesten, of anR-transient Markov chain on a countable state spaceS∪ {δ}, where δ is absorbing. The transition matrixKonSis irreducible and strictly substochastic. We determine the Yaglom limit, that is, the limiting conditional behavior given nonabsorption. Each starting statex∈Sresults in a different Yaglom limit. Each Yaglom limit is anR-1-invariant quasi-stationary distribution, whereRis the convergence parameter ofK. Yaglom limits that depend on the starting state are related to a nontrivialR-1-Martin boundary.