A martingale decomposition for quadratic forms of Markov chains (with applications)

作者：Atchade Yves F^*; Cattaneo Matias D

来源：Stochastic Processes and Their Applications, 2014, 124(1): 646-677.

DOI：10.1016/j.spa.2013.09.00

• Central limit theorems
• Markov chains
• Markov chain Monte Carlo
• Martingale approximations
• Quadratic forms
• U-statistics

We develop a martingale-based decomposition for a general class of quadratic forms of Markov chains, which resembles the well-known Hoeffding decomposition of U-statistics of i.i.d. data up to a reminder term. To illustrate the applicability of our results, we discuss how this decomposition may be used to studying the large-sample properties of certain statistics in two problems: (i) we examine the asymptotic behavior of lag-window estimators in time series, and (ii) we derive an asymptotic linear representation and limiting distribution of U-statistics with varying kernels in time series. We also discuss simplified examples of interest in statistics and econometrics.

• 出版日期2014-1