On clusters of high extremes of Gaussian stationary processes with ǫ-separation (original) (raw)
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Electronic Journal of Probability, 2010
The clustering of extremes values of a stationary Gaussian process X (t), t ∈ [0, T ] is considered, where at least two time points of extreme values above a high threshold are separated by at least a small positive value ǫ. Under certain assumptions on the correlation function of the process, the asymptotic behavior of the probability of such a pattern of clusters of exceedances is derived exactly where the level to be exceeded by the extreme values, tends to ∞. The excursion behaviour of the paths in such an event is almost deterministic and does not depend on the high level u. We discuss the pattern and the asymptotic probabilities of such clusters of exceedances.
Extremes of a certain class of Gaussian processes
Stochastic Processes and their Applications, 1999
We consider the extreme values of fractional Brownian motions, self-similar Gaussian processes and more general Gaussian processes which have a trend −ct ÿ for some constants c; ÿ ¿ 0 and a variance t 2H. We derive the tail behaviour of these extremes and show that they occur mainly in the neighbourhood of the unique point t0 where the related boundary function (u + ct ÿ)=t H is minimal. We consider the case that H ¡ ÿ.
Extremes of Gaussian Processes with Random Variance
Electronic Journal of Probability, 2011
Let ξ(t) be a standard locally stationary Gaussian process with covariance function 1 − r(t, t + s) ∼ C(t)|s| α as s → 0, with 0 < α ≤ 2 and C(t) a positive bounded continuous function. We are interested in the exceedance probabilities of ξ(t) with a random standard deviation η(t) = η − ζt β , where η and ζ are non-negative bounded random variables. We investigate the asymptotic behavior of the extreme values of the process ξ(t)η(t) under some specific conditions which depends on the relation between α and β.
Extremes of independent Gaussian processes
Extremes, 2011
For every n ∈ N, let X 1n , . . . , Xnn be independent copies of a zero-mean Gaussian process Xn = {Xn(t), t ∈ T }. We describe all processes which can be obtained as limits, as n → ∞, of the process an(Mn − bn), where Mn(t) = max i=1,...,n X in (t) and an, bn are normalizing constants. We also provide an analogous characterization for the limits of the process anLn, where Ln(t) = min i=1,...,n |X in (t)|.
Extremes of Gaussian Processes with Maximal Variance near the Boundary Points
Methodology And Computing In Applied Probability
Let Xt, t[0Y 1, be a Gaussian process with continuous paths with mean zero and nonconstant variance. The largest values of the Gaussian process occur in the neighborhood of the points of maximum variance. If there is a unique ®xed point t 0 in the interval 0Y 1, the behavior of Pfsup t[0Y1 Xt4ug is known for u??. We investigate the case where the unique point t 0 t u depends on u and tends to the boundary. This is reasonable for a family of Gaussian processes X u t depending on u, which have for each u such a unique point t u tending to the boundary as u??. We derive the asymptotic behavior of Pfsup t [ 0Y1 Xt4ug, depending on the rate as t u tends to 0 or 1. Some applications are mentioned and the computation of a particular case is used to compare simulated probabilities with the asymptotic formula. We consider the exceedances of such a nonconstant boundary by a Ornstein-Uhlenbeck process. It shows the dif®culties to simulate such rare events, when u is large.
On limiting cluster size distributions for processes of exceedances for stationary sequences
arXiv (Cornell University), 2010
It is well known that, under broad assumptions, the time-scaled point process of exceedances of a high level by a stationary sequence converges to a compound Poisson process as the level grows. The purpose of this note is to demonstrate that, for any given distribution G on N, there exists a stationary sequence for which the compounding law of this limiting process of exceedances will coincide with G.
Extreme values of a portfolio of Gaussian processes and a trend
Extremes, 2005
We consider the extreme values of a portfolio of independent continuous Gaussian processes P k i¼1 w i X i ðtÞ (w i 2 R; k 2 N) which are asymptotically locally stationary, with expectations E½X i ðtÞ ¼ 0 and variances Var½X i ðtÞ ¼ d i t 2H i ðd i 2 R þ ; 0 < H i < 1Þ, and a trend Àct for some constants ; c > 0 with > H i. We derive the probability Pfsup t>0 P k i¼1 w i X i ðtÞ À ct > ug for u ! 1, which may be interpreted as ruin probability.
On the exceedance point process for a stationary sequence
Probability Theory and Related Fields, 1988
It 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.