Extremes in Continuous Stochastic Processes

Abstract

The by now well developed extremal theory for stationary sequences, including the Extremal Types Theorem and criteria for convergence and domains of attraction, has interesting extensions to continuous parameter cases. One way of using discrete parameter results to obtain these extensions—this is the approach to be reviewed here—proceeds by the simple device of expressing the maximum over an expanding interval of length T = n, say, as the maximum of n “submaxima” over fixed intervals, viz M(T)=max{ζ₁,...,ζ_n} where ζ = sup{(t); i − 1 ≤ t <≤ i}. The proofs involve three main ingredients, (i) results on the tail of the distribution of one ζ i.e. of the maximum over a fixed interval, (ii) mixing conditions; which limits the dependence between extremes of widely separated Vs and (iii) clustering conditions; which specify the dependence between neighboring ζ’s. Each of (i)–(iii) also involves rather elaborate “discretization” procedures to enable probabilities to be calculated from finite-dimensional distributions. Methods and results for the Gaussian case will be stressed, following the extensive treatment in Part III of Leadbetter, Lindgren and Rootzén: Extremes and related properties of stationary sequences and processes, Springer (1983). Finally, alternative approaches will be briefly commented on.