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LIMIT THEOREMS FOR QUADRATIC FORMS OF MARKOV CHAINS YVES F. ATCHADE AND MATIAS D. CATTANEO
 

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 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.
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 time-dependent 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

  

Source: Atchadé, Yves F. - Department of Statistics, University of Michigan

 

Collections: Mathematics