Large deviation results for statistics of short-and long-memory Gaussian processes

This paper discusses the large deviation principle of several important statistics for short- and long-memory Gaussian processes. First, large deviation theorems for the log-likelihood ratio and quadratic forms for a short-memory Gaussian process with mean function are proved. Their asymptotics are described by the large deviation rate functions. Since they are complicated, they are numerically evaluated and illustrated using the Maple V system (Char et al., 1991a,b). Second, the large deviation theorem of the log-likelihood ratio statistic for a long-memory Gaussian process with constant mean is proved. The asymptotics of the long-memory case differ greatly from those of the short-memory case. The maximum likelihood estimator of a spectral parameter for a short-memory Gaussian stationary process is asymptotically efficient in the sense of Bahadur.