Estimation of First Returning Speed in Null Recurrent Continuous-Time Markov Chains

TL;DR

The paper advances a network-based framework to study null recurrence in continuous-time Markov chains by linking first returning times to electric-network quantities. It derives tail and finiteness criteria for $E^x(w(\tau_x))$ under slow-growing weights, and translates CTMC return problems into effective conductance calculations via $C_{x\leftrightarrow\Delta}(\lambda)$. The authors develop a general procedure for birth-death processes, providing explicit bounds and conditions, and apply the method to linear-growth and alternating-rate bilateral birth-death models, yielding precise thresholds for finiteness of moments of return times. This network-theoretic approach offers computable, versatile tools for classifying null-recurrent CTMCs and quantifying their first-return dynamics across key models.

Abstract

This paper establishes a novel connection between null-recurrent CTMCs and electric networks, offering a systematic classification of null-recurrent behavior based on the first returning speed. By leveraging techniques from electric network theory, we present a general method for estimating the first returning speed of null recurrent birth-death processes and provide some important examples.

Figures (6)

• Figure 1: Electric network on $\mathbf{N}$
• Figure 2: Short-circuiting electric network on $\mathbf{N}$
• Figure 3: Electric network for upper bound
• Figure 4: Open-circuiting electric network on $\mathbf{N}$
• Figure 5: Electric network for lower bound

Theorems & Definitions (11)

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