# Subgeometric Rates of Convergence for Discrete Time Markov Chains under Discrete Time Subordination

## TL;DR

This paper investigates how certain subgeometric convergence rates of Markov chains are preserved under discrete time subordination, providing new insights into their invariant measure convergence behavior.

## Contribution

It establishes conditions under which subgeometric convergence rates are inherited in discrete time Markov chains via subordination, extending continuous-time results.

## Key findings

- Identifies three typical subgeometric convergence rates.
- Provides moment estimates for discrete time subordinators.
- Shows inheritance of convergence rates under specific conditions.

## Abstract

In this note, we are concerned with the subgeometric rate of convergence of a Markov chain with discrete time parameter to its invariant measure in the $f$-norm. We clarify how three typical subgeometric rates of convergence are inherited under a discrete time version of Bochner's subordination. The crucial point is to establish the corresponding moment estimates for discrete time subordinators under some reasonable conditions on the underlying Bernstein function.