Extremes of multidimensional Gaussian processes

Extremes of Independent Gaussian Processes

For every n ∈ N, let X1n, . . . , 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) = maxi=1,...,n Xin(t) and an, bn are normalizing constants. We also provide an analogous characterization for the limits of the process anLn, where Ln(t) = mini=1,...,n |Xin(t)|.

On Extremes of Multidimensional Stationary Diffusion Processes in Euclidean Norm

Let (Xt)t≥0 be a R-valued stationary reversible diffusion process. We investigate the asymptotic behavior of MT := max0≤t≤T |Xt|, where | · | is the Euclidean norm in R. The aim of this paper is to characterize the tail asymptotics of MT for fixed T > 0 as well as the long time behavior of MT as T → ∞. This is related to spectral asymptotics of the generator of (Xt)t≥0 subject to Dirichlet boun...

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 2H ∈ (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 Yt − φ(t) > u) as u...

Multidimensional Wick-itô Formula for Gaussian Processes

Singular ergodic control for multidimensional Gaussian processes