Holger Rootzén

Basic Properties of Extremes and Level Crossings

We turn our attention now to continuous parameter stationary processes. We shall be especially concerned with stationary normal processes in this and most of the subsequent chapters but begin with a discussion of some basic properties which are relevant, whether or not the process is normal, and which will be useful in the discussion of extremal behaviour in later chapters.

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(un). Limits for probabilities P{M(T) ≤ uT} are then considered for arbitrary families of constants {uT}, leading, in particular, to a determination of domains of attraction.

Maxima of Stationary Sequences

In this chapter, we extend the classical extreme value theory of Chapter 1 to apply to a wide class of dependent (stationary) sequences. The stationary sequences involved will be those exhibiting a dependence structure which is not “too strong”. Specifically, a distributional type of mixing condition— weaker than the usual forms of dependence restriction such as strong mixing—will be used as a basic assumption in the development of the theory.

Sample Path Properties at Upcrossings

Our main concern in the previous chapter has been the numbers and locations of upcrossings of high levels, and the relations between the upcrossings of several adjacent levels.For instance, we know from Theorem 9.3.2 and relation (9.2.3) that for a standard normal process each upcrossing of the high level u = uτ; with a probability p = τ*/τ is accompanied by an upcrossing also of the level n n

Maxima and Minima and Extremal Theory for Dependent Processes

u_{tau * } = u - frac{{log p}} {u},

Maxima and Crossings of Nondifferentiable Normal Processes

n nasymptotically independently of all other upcrossings of u τ, and uτ*.

Exceedances of Levels and kth Largest Maxima

While our primary concern in this volume is with stationary processes, the results and methods may be used to apply simply to some important nonstationary cases. In particular, this is so for nonstationary normal sequences having a wide variety of possible mean and correlation structures, which is the situation considered first in this chapter.

Application of Extremes and Crossings under Dependence

Trivially, extremes in two or more mutually independent processes are independent. In this chapter we shall establish the perhaps somewhat surprising fact that, asymptotically, independence of extremes holds for normal processes even when they are highly correlated. However, we shall first consider the asymptotic independence of maxima and minima in one normal process. Since minima of ξ(t) are maxima for — ξ(t), this can in fact be regarded as a special case of independence between extremes in two processes, namely between the maxima in the completely dependent processes ξ(t)and-ξ(t).