ABSTRACT
This chapter discusses some limit theorems that are useful for statistical inference. There is an extensive literature on limit theorems for long-memory processes. The simplest time series models are Gaussian processes. Their distribution is fully determined by the expected value and the covariances. Methods of inference can therefore be obtained by restricting attention to the first two moments only. In general, it is unlikely that an observed time series is exactly Gaussian. Good statistical inference procedures should therefore remain valid approximately, even if the actual process deviates from this ideal model. Quadratic forms play a central role in approximations of the Gaussian maximum likelihood. The maximum likelihood estimator of T is obtained by maximizing g with respect to T. The limiting behavior of linear sums, quadratic forms, and higher order polynomial forms in G (Xt) turns out to be essentially characterized by the corresponding limiting behavior for Hermite polynomials Hk (Xt).
