Extreme Values, Regular Variation, and Point Processes.

1989 ◽  
Vol 84 (407) ◽  
pp. 845
Author(s):  
Alan Kimber ◽  
Sidney I. Resnick
Keyword(s):  
Regular Variation ◽  
Point Processes ◽  
Extreme Values

Point processes, regular variation and weak convergence

1986 ◽  
Vol 18 (01) ◽  
pp. 66-138 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Weak Convergence ◽  
Function Space ◽  
Regular Variation ◽  
Point Processes ◽  
Sufficient Condition ◽  
Space Setting

A method is reviewed for proving weak convergence in a function-space setting when regular variation is a sufficient condition. Point processes and weak convergence techniques involving continuity arguments play a central role. The method is dimensionless and holds computations to a minimum. Many applications of the methods to processes derived from sums and maxima are given.

Extreme values of independent stochastic processes

1977 ◽  
Vol 14 (4) ◽  
pp. 732-739 ◽  
Author(s):  
Bruce M. Brown ◽  
Sidney I. Resnick
Keyword(s):  
Stochastic Processes ◽  
Point Process ◽  
Time Scales ◽  
Poisson Point Process ◽  
Point Processes ◽  
Extreme Values ◽  
Weak Limit ◽  
One Dimensional ◽  
Limit Process ◽  
Related Point

The maxima of independent Weiner processes spatially normalized with time scales compressed is considered and it is shown that a weak limit process exists. This limit process is stationary, and its one-dimensional distributions are of standard extreme-value type. The method of proof involves showing convergence of related point processes to a limit Poisson point process. The method is extended to handle the maxima of independent Ornstein–Uhlenbeck processes.

scholarly journals On Heavy-tailed Crack Distribution for Loss Severity Modeling

2017 ◽  
Vol 6 (6) ◽  
pp. 92 ◽  
Author(s):  
Taehan Bae ◽  
Jingjiao Chen
Keyword(s):  
Regular Variation ◽  
Extreme Values ◽  
Model Fitting ◽  
Parametric Models ◽  
Data Sets ◽  
Crack Distribution ◽  
Loss Data ◽  
Catastrophic Loss ◽  
Heavy Tailed ◽  
Loss Severity

Heavy-tailedness and right-skewness are two typical features of loss data resulting from catastrophic events such as storms or earthquakes. In this paper we study the tail properties of the generalized crack distribution which has recently been introduced as an extension of the Birnbaum-Saunders distribution and the three-parameter Gaussian crack distribution. The theoretical tail relationships between the auxiliary (or baseline) distribution and the resulting generalized crack distribution are studied relying on the classical theories of extreme values and regular variation. A few concrete examples of heavy-tailed crack distribution are constructed and used for model fitting exercises on both simulated and real catastrophic loss data sets. The fitting results show that the heavy-tailed crack distribution with an appropriate choice of baseline density function outperforms some other commonly used parametric models.

scholarly journals Large Deviations for Point Processes Based on Stationary Sequences with Heavy Tails

2010 ◽  
Vol 47 (1) ◽  
pp. 1-40 ◽  
Author(s):  
Henrik Hult ◽  
Gennady Samorodnitsky
Keyword(s):  
Large Deviations ◽  
Point Processes ◽  
Heavy Tails ◽  
Extreme Values ◽  
Stationary Sequence ◽  
Random Coefficients ◽  
Ruin Probabilities ◽  
Stationary Sequences ◽  
Linear Processes ◽  
Infinite Linear

In this paper we propose a framework that facilitates the study of large deviations for point processes based on stationary sequences with regularly varying tails. This framework allows us to keep track both of the magnitude of the extreme values of a process and the order in which these extreme values appear. Particular emphasis is put on (infinite) linear processes with random coefficients. The proposed framework provides a fairly complete description of the joint asymptotic behavior of the large values of the stationary sequence. We apply the general result on large deviations for point processes to derive the asymptotic decay of certain probabilities related to partial sum processes as well as ruin probabilities.

scholarly journals Free point processes and free extreme values

2009 ◽  
Vol 147 (1-2) ◽  
pp. 161-183 ◽  
Author(s):  
G. Ben Arous ◽  
V. Kargin
Keyword(s):  
Point Processes ◽  
Extreme Values
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