M. R. Leadbetter

Extremes and local dependence in stationary sequences

SummaryExtensions of classical extreme value theory to apply to stationary sequences generally make use of two types of dependence restriction:(a)a weak “mixing condition” restricting long range dependence(b)a local condition restricting the “clustering” of high level exceedances. The purpose of this paper is to investigate extremal properties when the local condition (b) is omitted. It is found that, under general conditions, the type of the limiting distribution for maxima is unaltered. The precise modifications and the degree of clustering of high level exceedances are found to be largely described by a parameter here called the “extremal index” of the sequence.

Stationary and related stochastic processes : sample function properties and their applications

Book on stationary and related stochastic processes covering sample function properties and applications, Hilbert space geometry, etc

On the exceedance point process for a stationary sequence

SummaryIt is known that the exceedance points of a high level by a stationary sequence are asymptotically Poisson as the level increases, under appropriate long range and local dependence conditions. When the local dependence conditions are relaxed, clustering of exceedances may occur, based on Poisson positions for the clusters. In this paper a detailed analysis of the exceedance point process is given, and shows that, under wide conditions, any limiting point process for exceedances is necessarily compound Poisson. More generally the possible random measure limits for normalized exceedance point processes are characterized. Sufficient conditions are also given for the existence of a point process limit. The limiting distributions of extreme order statistics are derived as corollaries.

On extreme values in stationary sequences

SummaryIn this paper, extreme value theory is considered for stationary sequences ζn satisfying dependence restrictions significantly weaker than strong mixing. The aims of the paper are:(i)To prove the basic theorem of Gnedenko concerning the existence of three possible non-degenerate asymptotic forms for the distribution of the maximum Mn = max(ξ1...ξn), for such sequences.(ii)To obtain limiting laws of the form

On the Renewal Function for the Weibull Distribution

\mathop {\lim }\limits_{n \to \infty } \Pr \{ M_n^{(r)} \underset{\raise0.3em\hbox{

On high level exceedance modeling and tail inference

\smash{\scriptscriptstyle-}

On Extreme Values in Stationary Random Fields

}}{ \leqslant } u\} = e^{ - \tau } \sum\limits_{s = 0}^{r - 1} {\tau ^s /S!}

Conditions for the convergence in distribution of maxima of stationary normal processes

where Mn(r)is the r-th largest of ξ1...ξn, and Prξ1>un∼Τ/n. Poisson properties (akin to those known for the upcrossings of a high level by a stationary normal process) are developed and used to obtain these results.(iii)As a consequence of (ii), to show that the asymptotic distribution of Mn(r)(normalized) is the same as if the {ξn} were i.i.d.(iv)To show that the assumptions used are satisfied, in particular by stationary normal sequences, under mild covariance conditions.