Mixing conditions and limit theorems for maxima of some stationary sequences. Technical report No. 18

Abstract

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 foundmore » 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.« less