Maxima and Crossings of Nondifferentiable Normal Processes

Abstract

The basic assumption of the previous chapters has been that the covariance function r(T) of the stationary normal process £(r) has an expansion $r\left( \tau \right) = 1 - {\lambda _2}{\tau ^2}/2o\left( {{\tau ^2}} \right)$ as τ → 0. In this chapter we shall consider the more general class of conariances which have the expansion $r\left( \tau \right) = 1 - C\left| \tau \right|^\alpha + o\left( {\left| \tau \right|^\alpha } \right)$, where the positive constant a may be less than 2. This includes covariances of the form exp $\left( { - \left| \tau \right|^\alpha } \right)$, the case α = 1 being that of the Ornstein-Uhlenbeck process. Since the mean number of upcrossings of any level per unit time is infinite when α < 2, the methods of Chapter 8 do not apply in such cases. However, it will be shown by different methods that the double exponential limiting law for the maximum still applies with appropriately defined normalizing constants, if (8.1.2) (or a slightly weaker version) holds. This, of course, also provides an alternative derivation of the results of Chapter 8 when α = 2. Finally, while clearly no Poisson result is possible for upcrossings when α < 2, it will be seen that Poisson limits may be obtained for the related concept of ε-upcrossings, defined similarly to the e-maxima of Chapter 9.Chapter 13 Extremes of Continuous Parameter Stationary Processes Our primary task in this chapter will be to discuss continuous parameter analogues of the sequence results of Chapter 3, and, in particular, to obtain a corresponding version of the Extremal Types Theorem which applies in the continuous parameter case. This will be taken up in the first section, using a continuous parameter analogue of the dependence restriction D(u _n ). Limits for probabilities PM(T) ≤ u _T are then considered for arbitrary families of constants u_T, leading, in particular, to a determination of domains of attraction.