Point processes of exits by bivariate Gaussian processes and extremal theory for the chi^2-process and its concomitants

Let ζ(t), η(t) be continuously differentiable Gaussian processes with mean zero, unit variance, and common covariance function r(t), and such that ζ(t) and η(t) are independent for all t, and consider the movements of a particle with time-varying coordinates (ζ(t), η(t)). The time and location of the exists of the particle across a circle with radius u defines a point process in R3 with its points located on the cylinder {(t, u cos θ, u sin θ); t ≥ 0, 0 ≤ θ < 2π}. It is shown that if r(t) log t → 0 as t → ∞, the time and space-normalized point process of exits converges in distribution to a Poisson process on the unit cylinder. As a consequence one obtains the asymptotic distribution of the maximum of a χ2-process, χ2(t) = ζ2(t) + η2(t), P{sup0≤t≤T χ2(t) ≤ u2} → e−τ if T(−r″(0)/2π)1/2u × exp(−u2/2) → τ as T, u → ∞. Furthermore, it is shown that the points in R3 generated by the local ε-maxima of χ2(t) converges to a Poisson process in R3 with intensity measure (in cylindrical polar coordinates) (2πr2)−1 dt dθ dr. As a consequence one obtains the asymptotic extremal distribution for any function g(ζ(t), η(t)) which is “almost quadratic” in the sense that Image Full-size image has a limit g*(θ) as r → ∞. Then P{sup0≤t≤T g(ζ(t), η(t)) ≤ u2} → exp(−(τ/2π) ∫ θ = 02π e−g*(θ) dθ) if T(−r″(0)/2π)1/2u exp(−u2/2) → τ as T, u → ∞