A Central Limit Theorem for Stationary m-Dependent Sequences

This chapter deals with sequences of random variables that allow for a limited amount of dependence, and which are stationary. Stationarity of a sequence \(Y_1,Y_2,\ldots \) of random variables means that, for each \(j\geq 1\) and \(k geq 0\) , the distribution of the random vector \((Y_j,\ldots ,Y_{j+k})\) does not depend on j and thus is invariant to time shifts. In particular, all the \(Y_j\) have the same distribution. Moreover, we assume that, for some non-negative integer m, this sequence is m-dependent, which means that the \(\sigma \) -fields generated by \(Y_1,\ldots ,Y_s\) and by \(Y_{s+m+1},\ Y_{s+m+2},\ldots \) are independent for each \(s\geq 1\) . A rich collection of examples of an m-dependent stationary sequence is provided by functions of blocks of length \(m+1\) of a sequence of independent and identically distributed random variables. The main result of this chapter is a central limit theorem for stationary m-dependent sequences, which generalizes the Lindeberg-Lévy central limit theorem.