Core Concepts
The authors introduce a new notion of quasi-stationary distribution for reversible ergodic Markov chains without absorbing states. This generalization of the classical quasi-stationary distribution is characterized by an optimal strong stationary time, representing the "hitting time of the stationary distribution", and exhibits similar exponential behavior and geometric interpretation as the classical case.
Abstract
The authors consider a discrete-time reversible ergodic Markov chain (Xt)t≥0 with finite state space X. They introduce a new notion of quasi-stationary distribution that does not require the presence of any absorbing state.
The key insights are:
- The hitting time of the absorbing set is replaced by an optimal strong stationary time, representing the "hitting time of the stationary distribution".
- The new quasi-stationary distribution corresponds to the natural generalization of the Yaglom limit.
- The quasi-stationary distribution can be written in terms of the eigenvectors of the underlying Markov kernel, allowing for a geometric interpretation.
- The quasi-stationary distribution exhibits the usual exponential behavior that characterizes classical quasi-stationary distributions and metastable systems.
- The authors provide examples showing that the phenomenology is richer compared to the absorbing case.
- Counterexamples are presented, demonstrating that the assumption on the reversibility cannot be weakened in general.