Kesten's Yaglom limit counterexample

Abstract

We consider a positive matrix Q, with entries {1,2,...}. Kesten showed that if there exists a constant 1 ≤ L < infinity and sequences u1 < u 2 < ... and d1 < d 2 < ... such that Q(i, j) = 0 whenever i < ur < ur + L < j or i > dr + L > dr > j for some r, then if Q is also substochastic, it has the strong ratio limit property, that is, limn→infinity Qn+mi,j Qnk,l =R-mfim jfk ml for a suitable R and some R-1-harmonic function f and R-1-invariant measure mu. In this thesis, we delve into a counterexample of Kesten that shows that for a positive irreducible matrix Q, that is the transition probability matrix of a Markov chain {Xn}, the Yaglom limit theorem is not valid if we drop one of the restriction imposed above, even when Q has a minimal normalized quasi-stationary distribution. In the last section, we show the ratio limit property must also then fail.