Point processes of exits by bivariate Gaussian processes and extremal theory for the chi^2process and its concomitants
(1980) In Journal of Multivariate Analysis 10(2). p.181206 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) +... (More)
 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
Image
Fullsize 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 → ∞. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/1273174
 author
 Lindgren, Georg ^{LU}
 organization
 publishing date
 1980
 type
 Contribution to journal
 publication status
 published
 subject
 keywords
 Convergence of point processes, extremal theory, reliability, χ2process, crossings, maxima
 in
 Journal of Multivariate Analysis
 volume
 10
 issue
 2
 pages
 181  206
 publisher
 Academic Press
 external identifiers

 scopus:0009481528
 ISSN
 0047259X
 DOI
 10.1016/0047259X(80)900135
 language
 English
 LU publication?
 yes
 id
 ee07ff18320b4e45ad2e802c8d1b946e (old id 1273174)
 date added to LUP
 20160401 17:08:14
 date last changed
 20201122 03:13:40
@article{ee07ff18320b4e45ad2e802c8d1b946e, 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<br/><br> Image<br/><br> Fullsize image<br/><br> 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 → ∞.}, author = {Lindgren, Georg}, issn = {0047259X}, language = {eng}, number = {2}, pages = {181206}, publisher = {Academic Press}, series = {Journal of Multivariate Analysis}, title = {Point processes of exits by bivariate Gaussian processes and extremal theory for the chi^2process and its concomitants}, url = {http://dx.doi.org/10.1016/0047259X(80)900135}, doi = {10.1016/0047259X(80)900135}, volume = {10}, year = {1980}, }