# Notes on: Qin, Q., & Hobert, J. P. (2017): Asymptotically stable drift and minorization for markov chains with application to albert and chib's algorithm

## Table of Contents

## Notation

- samples size
- covariates
- denotes the data (i.e. responses and covariates)
- is a sequence of data
- denotes the corresponding probability measure
- denotes an
**intractable probability density function (pdf)** - with denotes the
**Markov transition function (Mtf)**of an irreducible, aperiodic, Harris recurrent Markov chain with invariant probability measure

## Definitions

- convergence complexity analysis
- ascertain how the convergence rate of a MCMC scales with sample size and/or number of covariates
- drift
- minorization
- d&m method
- refers to the method of using
*drift*and*minorization*

The chain is called **geometrically ergodic** if there exists and such that

where denotes the **total variation norm** and is the m-step Mtf.

The **geometric convergence rate** of a chain is given by

The chain is geometrically ergodic if and only if .

## Overview

- Attempts to answer question "What can we say about the convergence properties of the Markov chains as ?"