Representation minimizing the expected attraction time

Identify, for a given irreducible and positive recurrent Markov chain on a countable state space, the random dynamical system representation that minimizes the expected time to hit the random attractor when the initial state is distributed according to the stationary distribution π; equivalently, minimize ∑_{x∈X} π(x) E[T_A(ω, x)] over all representations.

Background

Theorem 5.1 shows that when the Markov chain is ergodic of degree 2, the average (under π) expected time to hit the attractor is finite. Nonetheless, Example constructions demonstrate this average can vary widely with the choice of RDS representation.

The authors therefore ask which representation achieves the minimal average hitting time, a question with direct relevance to perfect sampling methods such as coupling from the past.