[Paper Review] Limit Theorems for quadratic forms of Markov Chains

Yves F. Atchadé, Matias D. Cattaneo|arXiv (Cornell University)|Aug 13, 2011

Markov Chains and Monte Carlo Methods16 references3 citations

TL;DR

This paper develops a martingale approximation method for quadratic forms of Markov chains, enabling systematic analysis of their asymptotic behavior. It establishes central limit theorems for U-statistics with varying kernels and lag-window estimators in non-stationary and non-geometrically ergodic settings, offering new robust inference tools for Markov Chain Monte Carlo simulations.

ABSTRACT

We develop a martingale approximation approach to studying the limiting behavior of quadratic forms of Markov chains. We use the technique to examine the asymptotic behavior of lag-window estimators in time series and we apply the results to Markov Chain Monte Carlo simulation. As another illustration, we use the method to derive a central limit theorem for U-statistics with varying kernels.

Motivation & Objective

• To develop a general, systematic framework for analyzing the asymptotic behavior of quadratic forms of Markov chains.
• To extend central limit theorems for U-statistics with time-varying kernels beyond i.i.d. or stationary assumptions.
• To study the asymptotic properties of lag-window estimators of long-run variance in non-geometrically ergodic and non-stationary Markov chains.
• To provide a theoretical foundation for robust Monte Carlo confidence intervals in MCMC simulations.
• To establish fixed-b asymptotics for non-stationary Markov chains, extending classical inference frameworks.

Proposed method

• Uses a bivariate analog of the Poisson equation to construct a martingale approximation for quadratic forms of Markov chains.
• Applies the approximation to decompose lag-window estimators of long-run variance, revealing new insights into classical and fixed-b asymptotics.
• Employs weighted norms and functional analytic tools (e.g., $W$-norms, $\mathcal{L}_W$ spaces) to control dependence and moments.
• Derives a weak law of large numbers for Markov chains under moment and ergodicity conditions involving $V_1$ and $V_2$-norms.
• Uses the Poisson equation solution to express the original quadratic form as a sum of martingale differences plus remainder terms.
• Applies Cauchy-Schwarz and moment bounds to control the remainder terms and establish convergence in probability.

Experimental results

Research questions

• RQ1 How can martingale approximation be extended to quadratic forms of Markov chains to enable asymptotic analysis?
• RQ2 What are the asymptotic properties of lag-window estimators of long-run variance in non-geometrically ergodic Markov chains?
• RQ3 Can central limit theorems be established for U-statistics with time-varying kernels without assuming stationarity or mixing conditions?
• RQ4 How does the fixed-b asymptotic framework extend to non-stationary Markov chains?
• RQ5 What conditions ensure the consistency and asymptotic normality of lag-window estimators in general Markovian settings?

Key findings

• A central limit theorem is established for U-statistics with varying kernels under weaker conditions than previous results, without requiring mixing or stationarity.
• The paper proves consistency of lag-window estimators for non-geometrically ergodic Markov chains, extending results by Flegal and Jones (2010) and Atchade (2011).
• Fixed-b asymptotics are extended to non-stationary Markov chains, providing a robust framework for MCMC inference.
• The martingale approximation method successfully decomposes quadratic forms into a sum of martingale differences and a remainder, with the remainder converging to zero in probability under appropriate moment and ergodicity conditions.
• A weak law of large numbers is derived for weighted sums of Markov chains under $V$-geometric ergodicity and moment conditions, enabling convergence of normalized deviations.
• The method provides a unified approach to analyzing quadratic forms, with explicit bounds on remainder terms via $\overline{V}_2$-norms and $\overline{\mathcal{U}}_1$, $\overline{\mathcal{V}}_1$ functions.