# Multivariate initial sequence estimators in Markov chain Monte Carlo

## TL;DR

This paper introduces a new multivariate initial sequence estimator for Markov chain Monte Carlo that accurately estimates the covariance matrix of sample means, improving stability and validity over existing methods.

## Contribution

It develops a novel multivariate extension of Geyer's Monte Carlo error estimation method, providing asymptotically valid and stable covariance matrix estimates in MCMC.

## Key findings

- The proposed estimator is asymptotically valid.
- Simulation experiments show improved stability.
- Finite sample properties are favorable.

## Abstract

Markov chain Monte Carlo (MCMC) is a simulation method commonly used for estimating expectations with respect to a given distribution. We consider estimating the covariance matrix of the asymptotic multivariate normal distribution of a vector of sample means. Geyer (1992) developed a Monte Carlo error estimation method for estimating a univariate mean. We propose a novel multivariate version of Geyer's method that provides an asymptotically valid estimator for the covariance matrix and results in stable Monte Carlo estimates. The finite sample properties of the proposed method are investigated via simulation experiments.