Large extremes of Gaussian chaos processes

We study probabilities of large extremes of Gaussian chaos processes, that is, homogeneous functions of Gaussian vector processes. Important examples are products of Gaussian processes and quadratic forms of them. Exact asymptotic behaviors of the probabilities are found. To this aim, we use joint results of E. Hashorva, D. Korshunov and the author on Gaussian chaos, as well as a substantially modified asymptotical Double Sum Method.     

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.

     

Gaussian OS-positive processes

Summary Gaussian processes satisfying Osterwalder-Schrader positivity are studied. A representation of the (generalized) covariance function of an OS-positive process as the Laplace transform of an operator-valued probability measure is given. It is shown that every Gaussian OS-positive process has a unique Gaussian canonical Markov extension. An explicit application is made to the generalized free Euclidean fields.Partially supported by the National Science Foundation under grant MCS-76 06332     

Gaussian stochastic processes

The survey is devoted to works appearing in the last 3–5 years and pertaining mainly to local properties of the trajectories of Gaussian processes, the behavior of trajectories in the uniform metric, and properties of level sets. Some new results are also presented.     

Oscillation of Gaussian processes

The properties of the oscillations of Banach-valued Gaussian processes are investigated. The oscillations of several Gaussian sequences are computed. The obtained results are used for the investigation of the properties of the trajectories of one-dimensional Gaussian processes.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 177, pp. 92–97, 1989.     

Extremes of vector-valued Gaussian processes

The seminal papers of Pickands (Pickands, 1967; Pickands, 1969) paved the way for a systematic study of high exceedance probabilities of both stationary and non-stationary Gaussian processes. Yet, in the vector-valued setting, due to the lack of key tools including Slepian’s Lemma, there has not been any methodological development in the literature for the study of extremes of vector-valued Gaussian processes. In this contribution we develop the uniform double-sum method for the vector-valued setting, obtaining the exact asymptotics of the high exceedance probabilities for both stationary and n on-stationary Gaussian processes. We apply our findings to the operator fractional Brownian motion and Ornstein–Uhlenbeck process.     

Extremes of threshold-dependent Gaussian processes

In this paper,we are concerned with the asymptotic behavior,as u→∞,of P{sup_t∈|0,T|X_u(t)u},where X_u(t),t∈|0,T|,u0 is a family of centered Gaussian processes with continuous trajectories.A key application of our findings concerns P{sup_t∈|0,T|(X(t)+g(t))u},as u→∞,for X a centered Gaussian process and g some measurable trend function.Further applications include the approximation of both the ruin time and the ruin probability of the Brownian motion risk model with constant force of interest.     

Extremes of independent Gaussian processes

For every n ∈ ℕ, let X _1n ,..., X _nn be independent copies of a zero-mean Gaussian process X _n  = {X _n (t), t ∈ T}. We describe all processes which can be obtained as limits, as n→ ∞, of the process a _n (M _n  − b _n ), where M _n (t) =  max _{i = 1,...,n} X _in (t), and a _n , b _n are normalizing constants. We also provide an analogous characterization for the limits of the process a _n L _n , where L _n (t) =  min _{i = 1,...,n} |X _in (t)|.     

The asymptotic behavior of extrema of compound Cox processes with nonzero means

We give necessary and sufficient conditions for the weak convergence of one-dimensional distributions of extrema of compound Cox processes with jumps having finite variances and nonzero expectations. No moment-type restrictions are imposed on the controlling process. As corollaries we obtain criteria of the normal convergence of extrema of compound Cox processes with nonzero means. Convergence rate estimates are also presented. Supported by the Russian Foundation for Basic Research (grant Nos. 96-01-01919 and 97-01-00271), the Russian Humanitarian Sciences Foundation (grant No. 97-02-02235), and the Committee on Knowledge Extension Research (CKER) of the American Society of Actuaries. Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part I.     

Sudakov-type minoration for gaussian chaos processes

Consider a setA of symmetricn×n matricesa=(a _i,j) _i,j≤n . Consider an independent sequence (g _i) _i≤n of standard normal random variables, and letM=Esup_a∈A|Σ_i,j⪯na_i,jg_ig_j|. Denote byN ₂(A, α) (resp.N _t(A, α)) the smallest number of balls of radiusα for thel ₂ norm ofR ^{n 2} (resp. the operator norm) needed to coverA. Then for a universal constantK we haveα(logN ₂(A, α))^1/4≤KM. This inequality is best possible. We also show that forδ≥0, there exists a constantK(δ) such thatα(logN _t≤K(δ)M. Work partially supported by an N.S.F. grant.     

Some comparisons for Gaussian processes

Extensions and variants are given for the well-known comparison principle for Gaussian processes based on ordering by pairwise distance.

     

Gaussian processes with biconvex covariances

Let R(s, t) be a continuous, nonnegative, real valued function on a ≤ s ≤ t ≤ b. Suppose $\text{?R}\text{?s}\text{≥ 0}$, $\text{?R}\text{?t}\text{≤ 0}$, and $\text{?}{}^2\text{R}\text{?t}\text{?t ≤ 0}$ in the interior of the domain. Then the extension of R to a symmetric function on [a, b] × [a, b] is a covariance function. Such a covariance is called biconvex. Let X(t) be a Gaussian process with mean 0 and biconvex covariance. X has a representation as a sum of simple moving averages of white noises on the line and plane. The germ field of X at every point t is generated by X(t) alone. X is locally nondeterministic. Under an additional assumption involving the partial derivatives of R near the diagonal, the local time of the sample function exists and is jointly continuous almost surely, so that the sample function is nowhere differentiable.