LIMIT-THEOREMS FOR SAMPLE COVARIANCES OF STATIONARY LINEAR-PROCESSES WITH APPLICATIONS TO SEQUENTIAL ESTIMATION

For a stationary linear process X-t = mu + Sigma(j=0)(infinity)a(j) ep silon(t-j), where epsilon(t) are i.i.d. (0, sigma(2)), rate of converg ence and strong consistency of an estimate for the asymptotic variance tau(2) = sigma(2)(Sigma(j=0)(infinity)a(j))(2) are established under conditions: Sigma(j=0)(infinity)\a(j)\ < infinity and E\epsilon(1)\(2p ) < infinity p greater than or equal to 2. In addition, it is shown th at, with Sigma(j=0)(infinity)\a(j)\ < infinity and E\epsilon(1)\(2p) < infinity, p > 1, lim(n --> infinity)nE((X) over bar(Nn) - mu)(2) = ta u(2), where {N-n:n is an element of N} is sequence of F-k = sigma(epsi lon(u):u less than or equal to k) stopping times satisfying some condi tions. Applications are made to the problem of the sequential point an d fixed width confidence interval estimation of the mean mu.