Abstract:
To quote the economist John Maynard Keynes: “The long run is a misleading guide to current affairs. In the long run we are all dead.” It makes more sense to study the state of an evanescent system given it has not yet expired. For a substochastic Markov chain with kernel K on a state space S with killing this amounts to the study of of the Yaglom limit; that is the limiting probability the state at time n is y given the chain has not been absorbed; i.e. lim_{n\to\infty}K^n(x,y)/K^n(x,S).
We given an example where the Yaglom limit depends on the starting state x. We explain this phenomenon by showing that when the rho-Martin entrance boundary is non-trivial the Yaglom limit may depends on the starting state of the Markov chain. The proof involves an analysis of the space-time rho-Martin entrance boundary.