 
Summary: LIMIT THEOREMS FOR QUADRATIC FORMS OF MARKOV CHAINS
YVES F. ATCHAD´E AND MATIAS D. CATTANEO
(April 2011)
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 lagwindow 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 Ustatistics with varying kernels.
1. Introduction
This paper deals with quadratic forms of the type
Un(hn) =
n
=1 j=1
wn( , j)hn(X , Xj), n 1, (1)
for a stochastic process {Xn, n 0}, weight matrices wn : N × N R and symmetric
kernels hn : X × X R. Quadratic forms of possibly timedependent random variables
naturally arise in a variety of statistical and econometric problems, and their asymptotic
properties are of particular importance to develop asymptotically valid inference proce
dures.
For independent sequences {Xn, n 0}, the well known Hoeffding decomposition
