On the Sample Mean and Variance of a Long Memory Process

Some properties of the sample mean and sample variance of a long memory process are described. It is shown that for a particular class of long memory processes the asymptotic relative efficiency of the k -decimated sample mean xn (k) (formed by taking the mean of every k -th observation) with respect to the sample mean xn of all n observations is 1, but that the deficiency (as defined by Hodges and Lehmann) of xn(k) with respect to xn is infinite. It is also shown that the sample variance s; of a long memory process can be badly biased toward 0: for any integer N and every c > 0 there exists a long memory process with variance a 2 such that Es; < ca2 for all sample sizes n :$;N. These properties of the sample mean and variance are not shared by "ordinary" stationary processes such as ARMA processes.