A Transience Criterion for Uniformly Bounded Markov Chains with Asymptotically Zero Mean Drift

TL;DR

The paper addresses classification of discrete-time Markov chains on $\\mathbb{N}_0$ using mean drift analysis and introduces a new transience criterion for uniformly bounded chains with asymptotically zero drift that does not rely on second-moment drift information. The approach centers on Foster-Lyapunov criteria and a novel locally regular monotone drift condition, culminating in a Lyapunov function $f(i)=\\sum_{k=i}^\\infty \\gamma_k^2$ whose drift $\\Delta f(i)$ is shown to be nonpositive to imply transience. The main contributions include a practical transience criterion under four regularity conditions, a corollary for drift sequences $\\gamma_i= C i^{-\\alpha}(1+\\varepsilon_i)$ with $\\alpha\\in(1/2,1)$, and an explicit example where Lamperti-type second-moment criteria fail but the new criterion succeeds. Overall, the results extend transience analysis in the zero-drift regime and provide tools for constructing and validating transient behavior in uniformly bounded Markov chains.

Abstract

In this paper, we give an overview of mean drift conditions for the state-space classification of discrete-time Markov Chains and we present a new transience criterion for uniformly bounded Markov Chains with asymptotically zero drift. The criterion does not need a condition on the second-moment drifts and can be applied to certain chains for which other criteria fail.

Figures (2)

• Figure 1: State transition diagram of Example \ref{['sitstill']} for $k=1$
• Figure 2: State transition diagram of Example \ref{['lopsided']} for $m=1$ and $n=2$

Theorems & Definitions (32)

• Theorem 2.1: Foster foster
• Corollary 2.2
• proof
• Theorem 2.3: Fayolle-Malyshev-Menshikov fayolle
• Remark 2.4
• Theorem 2.5: Mertens et al. mertens
• Theorem 2.6: Mertens et al. mertens
• Definition 3.1
• Lemma 3.2: Pakes' Lemma pakes
• proof