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

Let [zeta](t), [eta](t) be continuously differentiable Gaussian processes with mean zero, unit variance, and common covariance function r(t), and such that [zeta](t) and [eta](t) are independent for all t, and consider the movements of a particle with time-varying coordinates ([zeta](t), [eta](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 [theta], u sin [theta]); t >= 0, 0 0 as t --> [infinity], 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 [chi]2-process, [chi]2(t) = [zeta]2(t) + [eta]2(t), P{sup0 e-[tau] if T(-r''(0)/2[pi])1/2u - exp(-u2/2) --> [tau] as T, u --> [infinity]. Furthermore, it is shown that the points in R3 generated by the local [epsilon]-maxima of [chi]2(t) converges to a Poisson process in R3 with intensity measure (in cylindrical polar coordinates) (2[pi]r2)-1 dt d[theta] dr. As a consequence one obtains the asymptotic extremal distribution for any function g([zeta](t), [eta](t)) which is "almost quadratic" in the sense that has a limit g*([theta]) as r --> [infinity]. Then P{sup0 exp(-([tau]/2[pi]) [integral operator] [theta] = 02[pi] e-g*([theta]) d[theta]) if T(-r''(0)/2[pi])1/2u exp(-u2/2) --> [tau] as T, u --> [infinity].