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Multivariate strong invariance principles in Markov chain Monte Carlo

Arka Banerjee, Dootika Vats

TL;DR

The paper advances multivariate strong invariance principles (SIP) for Markov chain Monte Carlo (MCMC) by leveraging wide-sense regeneration under an $l$-step minorization, removing the need for a 1-step minorization. It proves an explicit SIP with rate $\mathcal{O}(n^{\beta}\log n)$, where $\beta = \max\{1/(2+\delta), 1/(2p), 1/4\}$, for polynomially or geometrically ergodic Markov chains, and provides verifiable moment conditions involving $\tau_1$ and the regenerative sums $Z_1$. These results yield tighter, checkable conditions for the strong consistency of MCMC variance estimators, including the batch-means estimator and a regenerative estimator, via explicit bounds on the rate function $\kappa(n)$ and the asymptotic covariance matrix $\Sigma_f = \Sigma_Z/\mu$. The practical impact is improved variance estimation and output analysis for MCMC, with applicability to multivariate functionals and without relying on a 1-step minorization, thus broadening the class of chains where SIP-based analysis is provably valid.

Abstract

Strong invariance principles in Markov chain Monte Carlo are crucial to theoretically grounded output analysis. Using the wide-sense regenerative nature of the process, we obtain explicit bounds in the strong invariance converging rates for partial sums of multivariate ergodic Markov chains. Consequently, we present results on the existence of strong invariance principles for both polynomially and geometrically ergodic Markov chains without requiring a 1-step minorization condition. Our tight and explicit rates have a direct impact on output analysis, as it allows the verification of important conditions in the strong consistency of certain variance estimators.

Multivariate strong invariance principles in Markov chain Monte Carlo

TL;DR

The paper advances multivariate strong invariance principles (SIP) for Markov chain Monte Carlo (MCMC) by leveraging wide-sense regeneration under an -step minorization, removing the need for a 1-step minorization. It proves an explicit SIP with rate , where , for polynomially or geometrically ergodic Markov chains, and provides verifiable moment conditions involving and the regenerative sums . These results yield tighter, checkable conditions for the strong consistency of MCMC variance estimators, including the batch-means estimator and a regenerative estimator, via explicit bounds on the rate function and the asymptotic covariance matrix . The practical impact is improved variance estimation and output analysis for MCMC, with applicability to multivariate functionals and without relying on a 1-step minorization, thus broadening the class of chains where SIP-based analysis is provably valid.

Abstract

Strong invariance principles in Markov chain Monte Carlo are crucial to theoretically grounded output analysis. Using the wide-sense regenerative nature of the process, we obtain explicit bounds in the strong invariance converging rates for partial sums of multivariate ergodic Markov chains. Consequently, we present results on the existence of strong invariance principles for both polynomially and geometrically ergodic Markov chains without requiring a 1-step minorization condition. Our tight and explicit rates have a direct impact on output analysis, as it allows the verification of important conditions in the strong consistency of certain variance estimators.
Paper Structure (12 sections, 9 theorems, 66 equations)

This paper contains 12 sections, 9 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Definitions and main result
  3. Markov chains and wide-sense regeneration
  4. Main result
  5. Application to MCMC variance estimation
  6. Batch means estimator
  7. Regenerative estimator
  8. Acknowledgments
  9. Some preliminary results
  10. Proof of Theorem \ref{['Ths']}
  11. Proof of Theorem \ref{['thm:mcmc_sip']}
  12. Identifying wide-sense regenerations

Key Result

Theorem 1

Let $\{X_{t}\}_{t\geq 1}$ be a $\pi$-Harris ergodic Markov chain and thus eq:mino holds. Suppose Then, on a suitably rich probability space, one can construct $\{X_t\}_{t \geq 1}$ together with a $d$-dimensional standard Wiener process $\{ W(t): t \ge 0 \}$ such that for $\beta = \max\{ 1/{(2+\delta)}, 1/2p, 1/4 \}$, as $n \to \infty$

Theorems & Definitions (22)

  • Theorem 1
  • proof
  • Remark 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 2
  • proof
  • Remark 2
  • ...and 12 more