Point processes of exits by bivariate Gaussian processes and extremal theory for the chi^2process and its concomitants
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Abstract
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 timevarying 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 spacenormalized 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 χ2process, 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 εmaxima 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
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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 → ∞.
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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 → ∞.
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Research areas and keywords  Subject classification (UKÄ) – MANDATORY
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Original language  English 

Pages (fromto)  181206 
Journal  Journal of Multivariate Analysis 
Volume  10 
Issue number  2 
State  Published  1980 
Peerreviewed  Yes 