In this thesis some extreme value-limit theorems are obtained for specific classes of strictly stationary sequences. These results, along with results for associative processes provide the basic tools used to prove the limit theorems in the later chapters. A class of first-order exponential autoregressive processes is also studied. The processes are shown to satisfy the condition of Loynes' theorem. The strong mixing condition which in general would be difficult to check is found to be much easier to verify in this special case due to the Markov structure of the process. A general condition is given which implies strong mixing for Markov and p/sup th/ order autoregressive processes. This condition is easy to check for Markov processes. A class of uniform first-order autoregressive processes is studied. These processes are shown to satisfy Leadbetter's D(u/sub n/) condition but they fail to satisfy D'(u/sub n/) and so existing limit theorems cannot be applied. A new limit theorem is obtained for the maximum of such processes and it is seen to be a different result from what would have been obtained if D'(u/sub n/) held. This result shows that classical extreme value theory cannot be applied to all practical problems. It was found that even when stationarity and mixing conditions are assumed, the limit can differ from the independent case. It also shows that for some first-order autoregressive processes the limit distribution can depend on the autocorrelation at lag 1 whereas for processes satisfying Leadbetter's or Loynes' conditions the limit does not depend on the lag 1 autocorrelation. Hopefully, this type result will have application to air pollution problems.

This book has a dual purpose. One of these is to present material which selec tively will be appropriate for a quarter or semester course in time series analysis and which will cover both the finite parameter and spectral approach. The second object is the presentation of topics of current research interest and some open questions. I mention these now. In particular, there is a discussion in Chapter III of the types of limit theorems that will imply asymptotic nor mality for covariance estimates and smoothings of the periodogram. This dis cussion allows one to get results on the asymptotic distribution of finite para meter estimates that are broader than those usually given in the literature in Chapter IV. A derivation of the asymptotic distribution for spectral (second order) estimates is given under an assumption of strong mixing in Chapter V. A discussion of higher order cumulant spectra and their large sample properties under appropriate moment conditions follows in Chapter VI. Probability density, conditional probability density and regression estimates are considered in Chapter VII under conditions of short range dependence. Chapter VIII deals with a number of topics. At first estimates for the structure function of a large class of non-Gaussian linear processes are constructed. One can determine much more about this structure or transfer function in the non-Gaussian case than one can for Gaussian processes. In particular, one can determine almost all the phase information.

Classical Extreme Value Theory-the asymptotic distributional theory for maxima of independent, identically distributed random variables-may be regarded as roughly half a century old, even though its roots reach further back into mathematical antiquity. During this period of time it has found significant application-exemplified best perhaps by the book Statistics of Extremes by E. J. Gumbel-as well as a rather complete theoretical development. More recently, beginning with the work of G. S. Watson, S. M. Berman, R. M. Loynes, and H. Cramer, there has been a developing interest in the extension of the theory to include, first, dependent sequences and then continuous parameter stationary processes. The early activity proceeded in two directions-the extension of general theory to certain dependent sequences (e.g., Watson and Loynes), and the beginning of a detailed theory for stationary sequences (Berman) and continuous parameter processes (Cramer) in the normal case. In recent years both lines of development have been actively pursued.