Extremes Of Gaussian Processes Research Articles

Large deviations of bivariate Gaussian extrema

We establish sharp tail asymptotics for componentwise extreme values of bivariate Gaussian random vectors with arbitrary correlation between the components. We consider two scaling regimes for the tail event in which we demonstrate the existence of a restricted large deviations principle and identify the unique rate function associated with these asymptotics. Our results identify when the maxima of both coordinates are typically attained by two different versus the same index, and how this depends on the correlation between the coordinates of the bivariate Gaussian random vectors. Our results complement a growing body of work on the extremes of Gaussian processes. The results are also relevant for steady-state performance and simulation analysis of networks of infinite server queues.

Extremes of randomly scaled Gumbel risks

We investigate the product Y1Y2 of two independent positive risks Y1 and Y2. If Y1 has distribution in the Gumbel max-domain of attraction with some auxiliary function which is regularly varying at infinity and Y2 is bounded, then we show that Y1Y2 has also distribution in the Gumbel max-domain of attraction. If both Y1,Y2 have log-Weibullian or Weibullian tail behaviour, we prove that Y1Y2 has log-Weibullian or Weibullian asymptotic tail behaviour, respectively. We present here three theoretical applications concerned with a) the limit of point-wise maxima of randomly scaled Gaussian processes, b) extremes of Gaussian processes over random intervals, and c) the tail of supremum of iterated processes.

Extremes of Gaussian processes with smooth random expectation and smooth random variance

Let ξ(t), t ∈ [0, T],T > 0, be a Gaussian stationary process with expectation 0 and variance 1, and let η(t) and μ(t) be other sufficiently smooth random processes independent of ξ(t). In this paper, we obtain an asymptotic exact result for P(sup t∈[0,T](η(t)ξ(t) + μ(t)) > u) as u→∞.

Extremes of Gaussian processes with a smooth random trend

Let ?(t), t ? R, be a Gaussian zero mean stationary process, and ?(t) another random process, smooth enough, being independent of ?(t). We will consider the process ?(t) + ?(t) such that conditioned on ?(t) it is a Gaussian process. We want to establish an asymptotic exact result for P (t?[o,T] sup (?(t) + ?(t)) > u), as u ? ?, where T > 0.

Extremes of Gaussian processes with a smooth random variance

Let ξ ( t ) be a standard stationary Gaussian process with covariance function r ( t ) , and η ( t ) , another smooth random process. We consider the probabilities of exceedances of ξ ( t ) η ( t ) above a high level u occurring in an interval [ 0 , T ] with T > 0 . We present asymptotically exact results for the probability of such events under certain smoothness conditions of this process ξ ( t ) η ( t ) , which is called the random variance process. We derive also a large deviation result for a general class of conditional Gaussian processes X ( t ) given a random element Y .

Extremes of Gaussian Processes with Random Variance

Let $\xi(t)$ be a standard locally stationary Gaussian process with covariance function $1-r(t,t+s)\sim C(t)|s|^\alpha$ as $s\to0$, with $0<\alpha\leq 2$ and $C(t)$ a positive bounded continuous function. We are interested in the exceedance probabilities of $\xi(t)$ with a random standard deviation $\eta(t)=\eta-\zeta t^\beta$, where $\eta$ and $\zeta$ are non-negative bounded random variables. We investigate the asymptotic behavior of the extreme values of the process $\xi(t)\eta(t)$ under some specific conditions which depends on the relation between $\alpha$ and $\beta$.

Extremes of Gaussian processes over an infinite horizon

Consider a centered separable Gaussian process Y with a variance function that is regularly varying at infinity with index 2 H ∈ ( 0 , 2 ) . Let φ be a ‘drift’ function that is strictly increasing, regularly varying at infinity with index β > H , and vanishing at the origin. Motivated by queueing and risk models, we investigate the asymptotics for u → ∞ of the probability P ( sup t ⩾ 0 Y t - φ ( t ) > u ) as u → ∞ . To obtain the asymptotics, we tailor the celebrated double sum method to our general framework. Two different families of correlation structures are studied, leading to four qualitatively different types of asymptotic behavior. A generalized Pickands’ constant appears in one of these cases. Our results cover both processes with stationary increments (including Gaussian integrated processes) and self-similar processes.

Reduced-Load Equivalence for Queues with Gaussian Input

In this note, we consider a queue fed by a number of independent heterogeneous Gaussian sources. We study under what conditions a reduced load equivalence holds, i.e., when a subset of the sources becomes asymptotically dominant as the buffer size increases. For this, recent results on extremes of Gaussian processes [6] are combined with de Haan theory. We explain how the results of this note relate to square root insensitivity and moderately heavy tails.

Extremes of Gaussian Processes with Maximal Variance near the Boundary Points

Let X(t), t∈[0,1], 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 fixed point t0 in the interval [0,1], the behavior of P{supt∈[0,1] X(t)>u} is known for u→∞. We investigate the case where the unique point t0 = tu depends on u and tends to the boundary. This is reasonable for a family of Gaussian processes Xu(t) depending on u, which have for each u such a unique point tu tending to the boundary as u→∞. We derive the asymptotic behavior of P{supt∈[0,1] X(t)>u}, depending on the rate as tu 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 difficulties to simulate such rare events, when u is large.

Extremes of Gaussian processes, on results of Piterbarg and Seleznjev

Abstract For a particular sequence of Gaussian processes we consider the maximum Mn(T) up to time T and its limiting behaviour as T=T(n) and n converges to ∞. This sequence occurs in the approximation of the path of the continuous Gaussian process by broken lines. This limiting behaviour was analyzed by Piterbarg and Seleznjev assuming certain conditions. We improve their result assuming a weaker long-range dependence condition.

Extremes of Gaussian Processes with Bimodal Spectra

An approximate solution is developed for the extreme‐value distribution of a stationary Gaussian process with a spectral density function that exhibits two well‐separated modes. Processes of this type arise in the analysis of the combined dynamic response to two loads or to one load that excites two of a structure's modes of vibration. Spectral moments of each of the modes taken separately are used to characterize the process; two envelope processes are then used to approximate the extreme value distribution. The proposed distribution is compared with other approximations and with results from Monte Carlo simulations.

Correction: Correction to "Sojourns and Extremes of Gaussian Processes"

Correction: Correction to "Sojourns and Extremes of Gaussian Processes"

Correction Notes: Correction to "Sojourns and Extremes of Gaussian Processes"

Correction Notes: Correction to "Sojourns and Extremes of Gaussian Processes"

Sojourns and Extremes of Gaussian Processes

Let $X(t), 0 \leqq t \leqq 1$, be a real Gaussian process with mean 0 and continuous sample functions. For $u > 0$, form the process $u(X(t) - u)$. In this paper two related problems are studied. (i) Let $G$ be a nonnegative measurable function, and put $L = \int^1_0 G(u(X(t) - u)) dt$. For certain classes of processes $X$ and functions $G$, we find, for $u \rightarrow \infty$, the limiting conditional distribution of $L$ given that it is positive. (ii) For the same class of processes $X$, we find the asymptotic form of $P(\max_{\lbrack 0,1 \rbrack} X(t) > u)$ for $u \rightarrow \infty$. Finally, these results are extended to the process with the moving barrier, $X(t) - f(t)$, where $f$ is a continuous function.