7 - Doob’s h-Transform: Transition from Recurrent to Transient Chain and Vice Versa

Summary

Chapter 7 is the most conceptual part of the book. Our purpose here is to describe, without superfluous details, a change of measure strategy which allows us to transform a recurrent chain into a transient one, and vice versa. It is motivated by the exponential change of measure technique which goes back to Cramer. In the context of large deviations in collective risk theory, this technique allows us to transform a negatively drifted random walk into one with positive drift. Doob’s h-transform is the most natural substitute for an exponential change of measure in the context of Lamperti’s problem, that is, in the context of Markov chains with asymptotically zero drift.

Such transformations connect naturally previous chapters on asymptotic behaviour of transient chains with subsequent chapters, which are devoted to recurrent chains. A very important, in comparison with the classical Doob’s h-transform, the novelty consists in the fact that we use weight functions which are not necessarily harmonic, they are only asymptotically harmonic at infinity. The main challenge is to identify such functions under various drift scenarios.