scholarly journals Rare events, temporal dependence, and the extremal index

2006 ◽  
Vol 43 (2) ◽  
pp. 463-485 ◽  
Author(s):  
Johan Segers
Keyword(s):  
Rare Events ◽  
Random Variables ◽  
Value Theory ◽  
Extremal Index ◽  
Triangular Array ◽  
High Threshold ◽  
Special Role ◽  
Temporal Dependence ◽  
Direct Form ◽  
Abstract Probability

Classical extreme value theory for stationary sequences of random variables can to a large extent be paraphrased as the study of exceedances over a high threshold. A special role within the description of the temporal dependence between such exceedances is played by the extremal index. Parts of this theory can be generalized not only to random variables on an arbitrary state space hitting certain failure sets, but even to a triangular array of rare events on an abstract probability space. In the case of M4 (maxima of multivariate moving maxima) processes, the arguments take a simple and direct form.

scholarly journals Rare events, temporal dependence, and the extremal index

2006 ◽  
Vol 43 (02) ◽  
pp. 463-485
Author(s):  
Johan Segers
Keyword(s):  
Rare Events ◽  
Random Variables ◽  
Value Theory ◽  
Extremal Index ◽  
Triangular Array ◽  
High Threshold ◽  
Special Role ◽  
Temporal Dependence ◽  
Direct Form ◽  
Abstract Probability

Classical extreme value theory for stationary sequences of random variables can to a large extent be paraphrased as the study of exceedances over a high threshold. A special role within the description of the temporal dependence between such exceedances is played by the extremal index. Parts of this theory can be generalized not only to random variables on an arbitrary state space hitting certain failure sets, but even to a triangular array of rare events on an abstract probability space. In the case of M4 (maxima of multivariate moving maxima) processes, the arguments take a simple and direct form.

A note on the point processes of rare events

1996 ◽  
Vol 33 (03) ◽  
pp. 654-663 ◽  
Author(s):  
J. Hüsler ◽  
M. Schmidt
Keyword(s):  
Extreme Value Theory ◽  
Asymptotic Approximation ◽  
Point Processes ◽  
Rare Events ◽  
Random Sequence ◽  
Value Theory ◽  
Extreme Value ◽  
Triangular Array

We discuss the limits of point processes which are generated by a triangular array of rare events. Such point processes are motivated by the exceedances of a high boundary by a random sequence since exceedances are rare events in this case. This application relates the problem to extreme value theory from where the method is used to treat the asymptotic approximation of these point processes. The presented general approach extends, unifies and clarifies some of the various conditions used in the extreme value theory.

A note on the point processes of rare events

1996 ◽  
Vol 33 (3) ◽  
pp. 654-663 ◽  
Author(s):  
J. Hüsler ◽  
M. Schmidt
Keyword(s):  
Extreme Value Theory ◽  
Asymptotic Approximation ◽  
Point Processes ◽  
Rare Events ◽  
Random Sequence ◽  
Value Theory ◽  
Extreme Value ◽  
Triangular Array

We discuss the limits of point processes which are generated by a triangular array of rare events. Such point processes are motivated by the exceedances of a high boundary by a random sequence since exceedances are rare events in this case. This application relates the problem to extreme value theory from where the method is used to treat the asymptotic approximation of these point processes. The presented general approach extends, unifies and clarifies some of the various conditions used in the extreme value theory.

scholarly journals Modeling Best Practice Life Expectancy Using Gumbel Autoregressive Models

Risks ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 51
Author(s):  
Anthony Medford
Keyword(s):  
Time Series ◽  
Life Expectancy ◽  
Extreme Value Theory ◽  
Autoregressive Model ◽  
Best Practice ◽  
Gumbel Distribution ◽  
Value Theory ◽  
Model Diagnostics ◽  
Temporal Dependence ◽  
Simulation Results

Best practice life expectancy has recently been modeled using extreme value theory. In this paper we present the Gumbel autoregressive model of order one—Gumbel AR(1)—as an option for modeling best practice life expectancy. This class of model represents a neat and coherent framework for modeling time series extremes. The Gumbel distribution accounts for the extreme nature of best practice life expectancy, while the AR structure accounts for the temporal dependence in the time series. Model diagnostics and simulation results indicate that these models present a viable alternative to Gaussian AR(1) models when dealing with time series of extremes and merit further exploration.

scholarly journals Extreme value theory for piecewise contracting maps with randomly applied stochastic perturbations

2016 ◽  
Vol 16 (03) ◽  
pp. 1660015 ◽  
Author(s):  
Davide Faranda ◽  
Jorge Milhazes Freitas ◽  
Pierre Guiraud ◽  
Sandro Vaienti
Keyword(s):  
Extreme Value Theory ◽  
Stationary Process ◽  
Higher Dimensions ◽  
Value Theory ◽  
Extreme Value ◽  
Extremal Index ◽  
Limiting Distributions ◽  
Stochastic Perturbations

We consider globally invertible and piecewise contracting maps in higher dimensions and perturb them with a particular kind of noise introduced by Lasota and Mackey. We got random transformations which are given by a stationary process: in this framework we develop an extreme value theory for a few classes of observables and we show how to get the (usual) limiting distributions together with an extremal index depending on the strength of the noise.

scholarly journals Limit theorems for triangular arrays under a relaxed asymptotic negligibility condition

1987 ◽  
Vol 42 (1) ◽  
pp. 117-128 ◽  
Author(s):  
Miloslav Jirina
Keyword(s):  
Asymptotic Behavior ◽  
Classical Theory ◽  
Limit Theorems ◽  
Random Variables ◽  
Triangular Array ◽  
Independent Random Variables ◽  
Triangular Arrays

AbstractLet {Xnk} be a triangular array of independent random variables satisfying the so-called tail-negligibility condition, i.e. such that Prob{|Xnk| > a} → 0 as both k, n → ∞. It is also assumed that for each fixed k, Xnk converges in distribution as n → ∞. Theorems on the asymptotic behavior of the row sums of the array, analogous to those of the classical theory under the uniform negligibility condition, are presented.

scholarly journals Two-faced families of non-commutative random variables having bi-free infinitely divisible distributions

2016 ◽  
Vol 27 (04) ◽  
pp. 1650037 ◽  
Author(s):  
Mingchu Gao
Keyword(s):  
Limit Theorem ◽  
Fock Space ◽  
Random Variables ◽  
Infinite Divisibility ◽  
Triangular Array ◽  
Divisible Distribution ◽  
Infinitely Divisible Distributions ◽  
Space Operator ◽  
Infinitely Divisible ◽  
Operator Model

We study two-faced families of non-commutative random variables having bi-free (additive) infinitely divisible distributions. We prove a limit theorem of the sums of bi-free two-faced families of random variables within a triangular array. As a corollary of our limit theorem, we get Voiculescu’s bi-free central limit theorem. Using the full Fock space operator model, we show that a two-faced pair of random variables has a bi-free (additive) infinitely divisible distribution if and only if its distribution is the limit distribution in our limit theorem. Finally, we characterize the bi-free (additive) infinite divisibility of the distribution of a two-faced pair of random variables in terms of bi-free Levy processes.

scholarly journals A counterexample concerning the extremal index

1988 ◽  
Vol 20 (3) ◽  
pp. 681-683 ◽  
Author(s):  
Richard L. Smith
Keyword(s):  
Cluster Size ◽  
Extremal Index ◽  
Stationary Sequence ◽  
High Threshold ◽  
Local Dependence

The concept of an extremal index, which is a measure of local dependence amongst the exceedances over a high threshold by a stationary sequence, has a natural interpretation as the reciprocal of mean cluster size. We exhibit a counterexample which shows that this interpretation is not necessarily correct.

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