http://library.um.ac.id/free-contents/downloadpdf.php/buku/laws-of-small-numbers-extremes-and-rare-events-michael-falk-jurg-husler-rolf-dieter-reiss-42718.pdf

Laws of small numbers: extremes and rare events / Michael Falk, Jurg Husler, Rolf-Dieter Reiss

In a variety of practical problems involving random processes, it is necessary to have statistical information on their level-crossing properties. This paper presents a survey of known results on certain aspects of this problem and provides a basis for further study in the area. The goal has been to give a broad view of the problems considered in the literature and a brief indication of the techniques used in their solution. Much material of a more or less historical nature has been included since, to the authors' knowledge, no other survey of this nature exists.

Level-crossing problems for random processes

: The purpose of this paper is to provide an overview of the asymptotic distributional theory of extreme values for a wide class of dependent stochastic sequences and continuous parameter processes. The theory contains the standard classical extreme value results for maxima and extreme order statistics as special cases but is richer on account of the diverse behavior possible under dependence in both discrete and continuous time contexts. Emphasis is placed on stationary cases but other important classes (e.g. Mark of sequences) are included. Significant ideas and methods are described rather than details, and in particular the nature and role of important underlying point processes (such as exceedances and upcrossings) are emphasized. Applications are given to particular classes of process (e.g. normal, moving average) and connections with related theory (such as convergence of sums) are indicated.

/pdf/external-theory-for-stochastic-processes-ojd30t0xp1.pdf

External Theory for Stochastic Processes.

We consider density estimates of the usual type generated by a weight function. Limt theorems are obtained for the maximum of the normalized deviation of the estimate from its expected value, and for quadratic norms of the same quantity. Using these results we study the behavior of tests of goodness-of-fit and confidence regions based on these statistics. In particular, we obtain a procedure which uniformly improves the chi-square goodness-of-fit test when the number of observations and cells is large and yet remains insensitive to the estimation of nuisance parameters. A new limit theorem for the maximum absolute value of a type of nonstationary Gaussian process is also proved.

/pdf/on-some-global-measures-of-the-deviations-of-density-2k83ayeluy.pdf

On Some Global Measures of the Deviations of Density Function Estimates

Extremes of Gaussian processes over an infinite horizon

: The authors consider two problems for separable mean zero Gaussian processes X(t) with correlation functions rho(t,s) for which 1-rho(t,s) is asymptotic to a regularly varying (at zero) function of /t-s/ with exponent 0=or < alpha =or <2. In showing the existence of such (stationary) processes for 0 = or < alpha < 2, the authors relate the magnitude of the tails of the spectral distributionsto the behavior of the covariance function at the origin. For 0 < alpha = or < 2, the authors obtain the asymptotic distribution of the maximum of X(t). This second result is used to obtain a result for X(t) as t approaches infinity similar to the 'so called' law of the iterated logarithm. (Author)

/pdf/asymptotic-properties-of-gaussian-processes-2vji8ixt44.pdf

Asymptotic Properties of Gaussian Processes

: The paper presents a result which generalizes the paper of Watanabe (Trans. AMS 148, 233-248). The result is extended also to the nonstationary process treated by Watanabe.

/pdf/an-asymptotic-0-1-behavior-of-gaussian-processes-q182w25f6n.pdf

An Asymptotic 0-1 Behavior of Gaussian Processes

/pdf/on-a-limit-distribution-of-high-level-crossings-of-a-2fue5apid1.pdf

On a Limit Distribution of High Level Crossings of a Stationary Gaussian Process

: It is shown that a stationary Gaussian process X(t), defined on t = 1,2, attains values which exceed any given non-decreasing function f(t) infinitely often with probability zero or one. The only assumption made is that the covariance function r(t) go to zero as t goes to infinity. When r(t) is of smaller order than t sup gamma for some gamma > 0, a test is given which distinguishes between the two cases of probability zero and one. Similar results are indicated when, instead, the index t assumes all the values in the interval (0, infinity) and the sample paths of x(t) are continuous. (Author)

Clifford Qualls

Papers

Asymptotic Properties of Gaussian Processes

An Asymptotic 0-1 Behavior of Gaussian Processes

On a Limit Distribution of High Level Crossings of a Stationary Gaussian Process

A Note on a O-1 Law for Stationary Gaussian Processes,