Extremes and Related Properties of Random Sequences and Processes

This chapter is primarily concerned with the central result of classical extreme value theory—the Extremal Types Theorem—which specifies the possible forms for the limiting distribution of maxima in sequences of i.i.d. random variables. In the derivation, the possible limiting distributions are identified with a class having a certain stability property—the so-called max-stable distributions. It is further shown that this class consists precisely of the three families known (loosely) as the three extreme value distributions.

In this chapter, we investigate properties of the exceedances of levels {u _n } by ξ_i, ξ₂,…, i.e. the points i for which ξ_i, > u_n, and as consequences, obtain limiting distributional results for the kth largest value among ξ₁,…, ξ₁. In particular, when the familiar assumption \(n\left( {1 - F\left( {u_n } \right)} \right) \to \tau \left( {0 < \tau < \infty } \right)\) holds (Equation (1.5.1)), it will be shown that the exceedances take on a Poisson character as n becomes large. This leads to the limiting distributions for the kth largest values for any fixed rank k = 1, 2,… (the kth “extreme order statistics”) and to their limiting joint distributions.

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.

Normality occupies a central place in probability and statistical theory, and a most important class of stationary sequences consists of those which are normal. Their importance is enhanced by the fact that their joint normal distributions are determined by the mean and the covariance structure of the sequence. In this chapter we investigate the extremal properties of stationary normal sequences. In particular covariance conditions will be obtained for the convergence of maxima to a Type I limit, both directly and by application of the general theory of Chapter 3.

In this chapter we return to the general situation and notation of Chapter 3 and consider the points; (regarded as “time instants”) at which the general stationary sequence {ξ_j} exceeds some given level u. These times of exceed-ance are stochastic in nature and may be viewed as a point process. Since exceedances of very high levels will be rare, one may suspect that this point process will take on a Poisson character at such levels. An explicit theorem along these lines will be proved and the asymptotic distributions of kth largest values (order statistics) obtained as corollaries. Generalizations of this theorem yield further results concerning joint distributions of kth largest values. The formal definition and simple properties of point processes which will be needed are given in the appendix.

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.

In this chapter the theory of maxima of mean square differentiable stationary normal processes will be developed under simple conditions—giving analogous results to those of Chapter 4. This will be approached using the properties of upcrossings developed in the previous chapter and will result in the limiting double exponential distribution for the maximum, with the appropriate scale and location normalization similar to that in Chapter 4.

In the limit theory for the maximum of a stationary normal process ξ(t), as developed in Chapter 8, substantial use was made of upcrossings, and of the obvious fact that the maximum exceeds u if there is at least one upcrossing of the level u. However, the upcrossings have an interest in their own right, and as we shall see here, they also contain considerable information about the local structure of the process. This chapter is devoted to the asymptotic Poisson character of the point process of upcrossings of increasingly high levels, and of the point process formed by the local maxima of the process.

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 = u_τ; with a probability p = τ*/τ is accompanied by an upcrossing also of the level asymptotically independently of all other upcrossings of u _τ, and u_τ*.

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).

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.

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

Extreme value distributions have found widespread use for the description of strength of materials and mechanical structures, often in combination with stochastic models for the loads and forces acting on the material. Thus it is often assumed that the maximum of several loads follows one of the extreme value distributions for maxima. More important, and also less obvious, is that the strength of a piece of material, such as a strip of paper or glass fibre, is sometimes determined by the strength of its weakest part, and then perhaps follows one of the extreme value distributions for minima. Based on this so-called weakest link principle much of the work has been directed towards a study of size effects in the testing of materials. By this we mean the empirical fact that the strength of a piece of material varies with its dimensions in a way which is typical for the type of material and the geometrical form of the object. An early attempt towards a statistical theory for this was made more than a century ago by Chaplin (1880, 1882); see also Lieblein (1954) and Harter (1977).

In this chapter we shall present some examples of continuous parameter processes and sequences with dependence where extreme value theory may be applied for descriptive or predictive purposes.