On the Class of Nilpotent Markov Chains, I. The Spectrum of Covariance Operator

#### A.M. Alhakim, J. Kawczak, S.A. Molchanov

2004, v.10, Issue 4, 629-652

ABSTRACT

We study the central limit theorem and the structure of the corresponding covariance operator for the Markov chains generated by successive (overlapping) $k$-tuples $(X_{n+1},\ldots,X_{n+k})$, $n=0,1,\ldots$ formed from the i.i.d.r.v. $\{X_n\}$. The potential application of the theory includes the design of statistical tests. In particular, we present the explicit spectral analysis of the covariance matrices related to Marsaglia's $k$ permutation test for $k=2, 3, 4, 5$.

Keywords: CLT for the nilpotent Markov chain,spectral decomposition,testing random number generators