Point processes of exits by bivariate Gaussian processes and extremal theory for the chi^2-process and its concomitants
(1980) In Journal of Multivariate Analysis 10(2). p.181-206- 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 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) +... (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 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 → ∞. (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, χ2-process, crossings, maxima
- in
- Journal of Multivariate Analysis
- volume
- 10
- issue
- 2
- pages
- 181 - 206
- publisher
- Academic Press
- external identifiers
-
- scopus:0009481528
- ISSN
- 0047-259X
- DOI
- 10.1016/0047-259X(80)90013-5
- language
- English
- LU publication?
- yes
- id
- ee07ff18-320b-4e45-ad2e-802c8d1b946e (old id 1273174)
- date added to LUP
- 2016-04-01 17:08:14
- date last changed
- 2021-08-15 03:12:03
@article{ee07ff18-320b-4e45-ad2e-802c8d1b946e, 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 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<br/><br> Image<br/><br> Full-size 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 = {{0047-259X}}, keywords = {{Convergence of point processes; extremal theory; reliability; χ2-process; crossings; maxima}}, language = {{eng}}, number = {{2}}, pages = {{181--206}}, publisher = {{Academic Press}}, series = {{Journal of Multivariate Analysis}}, title = {{Point processes of exits by bivariate Gaussian processes and extremal theory for the chi^2-process and its concomitants}}, url = {{http://dx.doi.org/10.1016/0047-259X(80)90013-5}}, doi = {{10.1016/0047-259X(80)90013-5}}, volume = {{10}}, year = {{1980}}, }