[Paper Review] Yaglom limits can depend on the starting state

Robert D. Foley, David McDonald|arXiv (Cornell University)|Sep 22, 2017

Stochastic processes and statistical mechanics17 references3 citations

TL;DR

This paper constructs a simple, irreducible, aperiodic, strictly substochastic Markov chain on a countable state space where the Yaglom limit—defined as the limiting conditional distribution given non-absorption—depends explicitly on the starting state. The key contribution is proving that such state-dependent Yaglom limits exist and are linked to a nontrivial $ R^{-1} $-Martin boundary, resolving an open question about whether Yaglom limits must be unique across initial states.

ABSTRACT

We construct a simple example, surely known to Harry Kesten, of an R-transient Markov chain on a countable state space S with cemetery state delta. The transition matrix K on S is irreducible and strictly substochastic. We determine the Yaglom limit, that is, the limiting conditional behavior given non-absorption. Each starting state x in S results in a different Yaglom limit. Each Yaglom limit is an rho-invariant quasi-stationary distribution where rho=1/R and R is the convergence parameter of $K$. Yaglom limits that depend on the starting state are related to a nontrivial rho-Martin entrance boundary.

Motivation & Objective

• To resolve an open question in quasi-stationary distribution theory: whether Yaglom limits must be independent of the starting state in irreducible, aperiodic, strictly substochastic Markov chains.
• To construct a concrete example of such a chain where each starting state $ x otin \{\delta\} $ yields a distinct Yaglom limit $ \pi_x $.
• To establish that the dependence of Yaglom limits on the initial state is tied to the structure of the $ R^{-1} $-Martin boundary, particularly when it is nontrivial.
• To demonstrate that the generalized strong ratio limit property (GSRLP) holds even when Yaglom limits differ across initial states, extending beyond Kesten’s SRLP framework.

Proposed method

• Construct a Markov chain on $ S \cup \{\delta\} $, where $ S $ is countably infinite, $ \delta $ is absorbing, and the transition matrix $ K $ on $ S $ is irreducible, aperiodic, and strictly substochastic.
• Define the Yaglom limit as $ \pi_x(y) = \lim_{n \to \infty} \mathbb{P}_x(X_n = y \mid X_n \in S) $, explicitly allowing dependence on the starting state $ x $.
• Use the duality between $ t $-invariant measures and $ t $-harmonic functions, particularly through Doob’s $ h $-transform and time reversal, to analyze the structure of invariant measures and harmonic functions.
• Characterize the $ R^{-1} $-Martin boundary and show that nontriviality of this boundary corresponds to the existence of multiple distinct Yaglom limits.
• Apply the Poisson-Martin integral representation to link $ R^{-1} $-invariant quasi-stationary distributions to the harmonic functions derived from minimal $ R^{-1} $-invariant measures.
• Use the 'hub-and-two-spoke' example to concretely illustrate the construction and verify that each starting state leads to a different limiting conditional distribution.

Experimental results

Research questions

• RQ1 Can Yaglom limits in irreducible, aperiodic, strictly substochastic Markov chains depend on the initial state?
• RQ2 What structural condition on the state space or transition kernel leads to state-dependent Yaglom limits?
• RQ3 How is the nontriviality of the $ R^{-1} $-Martin boundary related to the existence of multiple Yaglom limits?
• RQ4 Does the generalized strong ratio limit property (GSRLP) still hold when Yaglom limits differ across initial states?
• RQ5 Can a Yaglom limit fail to exist even when a $ \rho $-invariant quasi-stationary distribution exists?

Key findings

• The Yaglom limit is not unique across initial states: each starting state $ x \in S $ yields a distinct limiting conditional distribution $ \pi_x $.
• The existence of distinct Yaglom limits is directly linked to a nontrivial $ R^{-1} $-Martin boundary, where $ R $ is the convergence parameter of the transition matrix $ K $.
• Each Yaglom limit $ \pi_x $ is an $ R^{-1} $-invariant quasi-stationary distribution, meaning $ \pi_x K = R^{-1} \pi_x $.
• The generalized strong ratio limit property (GSRLP) holds even when Yaglom limits depend on the starting state, though the classical SRLP may fail.
• The constructed example is aperiodic and irreducible, yet still exhibits state-dependent Yaglom limits, contradicting the intuition that such limits should be unique.
• The hub-and-two-spoke example explicitly demonstrates that different starting states lead to different $ \pi_x $, with each $ \pi_x $ being an $ R^{-1} $-invariant measure on $ S $.