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Some properties of a normal process near a local maximum

Lindgren, Georg LU (1970) In Annals of Mathematical Statistics 41(6). p.1870-1883
Abstract
Consider a stationary normal process ξ(t) with mean zero and the covariance function r(t). Properties of the sample functions in the neighborhood of zeros, upcrossings of very high levels, etc. have been studied by, among others, Kac and Slepian, 1959 [4] and Slepian, 1962 [11]. In this paper we shall study the sample functions near local maxima of height u, especially as u → -∞, and mainly use similar methods as [4] and [11]. Then it is necessary to analyse carefully what is meant by "near a maximum of height u." In Section 2 we derive the "ergodic" definition, i.e. the definition which is possible to interpret by the aid of relative frequencies in a single realisation. This definition has been treated previously by Leadbetter, 1966 [5],... (More)
Consider a stationary normal process ξ(t) with mean zero and the covariance function r(t). Properties of the sample functions in the neighborhood of zeros, upcrossings of very high levels, etc. have been studied by, among others, Kac and Slepian, 1959 [4] and Slepian, 1962 [11]. In this paper we shall study the sample functions near local maxima of height u, especially as u → -∞, and mainly use similar methods as [4] and [11]. Then it is necessary to analyse carefully what is meant by "near a maximum of height u." In Section 2 we derive the "ergodic" definition, i.e. the definition which is possible to interpret by the aid of relative frequencies in a single realisation. This definition has been treated previously by Leadbetter, 1966 [5], and it turns out to be related to Kac and Slepian's horizontal window definition. In Section 3 we give a representation of ξ(t) near a maximum as the difference between a non-stationary normal process and a deterministic process, and in Section 4 we examine these processes as u → -∞. We have then to distinguish between two cases. A: Regular case. r(t) = 1 -λ<sub>2t</sub><sup>2</sup>/2 + λ<sub>4</sub> t<sup>4</sup>/4! - λ<sub>6</sub> t<sup>6</sup>/6! + o(t<sup>6</sup>) as t → 0, where the positive λ<sub>2k</sub> are the spectral moments. Then it is proved that if ξ(t) has a maximum of height u at t = 0 then, as u → -∞, egin{align*} (lambda_2lambda_6 - lambda_4^2)(lambda_4 - lambda_2^2)^{-1}{xi((lambda_2lambda_6 - lambda_4^2)^{-frac{1}{2}}(lambda_4 - lambda_2^2)^{frac{1}{2}}t|u|^{-1}) - u} \ sim |u|^{-3}{t^4/4! + omega(lambda_4 - lambda_2^2)^{frac{1}{2}}lambda_2^ {-frac{1}{2}}t^3/3! - zeta(lambda_4 - lambda_2^2)lambda_2 ^{-1}t^2/2}end{align*} where ω and ζ are independent random variables (rv), ω has a standard normal distribution and ζ has the density <latex>z exp (-z), z > 0</latex>. Thus, in the neighborhood of a very low maximum the sample functions are fourth degree polynomials with positive t<sup>4</sup>-term, symmetrically distributed t<sup>3</sup>-term, and a negatively distributed t<sup>2</sup>-term but without t-term. B: Irregular case. r(t) = 1 - λ<sub>2t</sub><sup>2</sup>/2 + λ<sub>4t</sub><sup>4</sup>/4! - λ<sub>5</sub>|t|<sup>5</sup>/5! + o(t<sup>5</sup>) as t → 0, where <latex>λ5 > 0</latex>. Now ξ(tu<sup>-2</sup>) - u ∼ |u|<sup>-5</sup>{λ<sub>2</sub>λ<sub>5</sub>(λ<sub>4</sub> - λ<sub>2</sub><sup>2</sup>)<sup>-1</sup> |t|<sup>3</sup>/3! + (2λ<sub>5</sub>)<sup>1/2</sup> ω(t) - ζ(λ<sub>4</sub> - λ<sub>2</sub><sup>2</sup>)λ<sub>2</sub> <sup>-1</sup>t<sup>2</sup>/2} where ω(t) is a non-stationary normal process whose second derivative is a Wiener process, independent of ζ which has the density <latex>z exp (-z), z > 0</latex>. The term λ<sub>5</sub>|t|<sup>5</sup>/5! "disturbs" the process in such a way that the order of the distance which can be surveyed is reduced from 1/|u| (in Case A) to 1/|u|<sup>2</sup>. The results are used in Section 5 to examine the distribution of the wave-length and the crest-to-trough wave-height, i.e., the amplitude, discussed by, among others, Cartwright and Longuet-Higgins, 1956 [1]. One hypothesis, sometimes found in the literature, [10], states that the amplitude has a Rayleigh distribution and is independent of the mean level. According to this hypothesis the amplitude is of the order 1/|u| as u → -∞ while the results of this paper show that it is of the order 1/|u|<sup>3</sup>. (Less)
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Annals of Mathematical Statistics
volume
41
issue
6
pages
1870 - 1883
publisher
Institute of Mathematical Statistics
ISSN
0003-4851
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English
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yes
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b77924e6-704e-449b-885f-20500a9c442e (old id 1273109)
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2009-06-03 17:13:43
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@article{b77924e6-704e-449b-885f-20500a9c442e,
  abstract     = {Consider a stationary normal process ξ(t) with mean zero and the covariance function r(t). Properties of the sample functions in the neighborhood of zeros, upcrossings of very high levels, etc. have been studied by, among others, Kac and Slepian, 1959 [4] and Slepian, 1962 [11]. In this paper we shall study the sample functions near local maxima of height u, especially as u → -∞, and mainly use similar methods as [4] and [11]. Then it is necessary to analyse carefully what is meant by "near a maximum of height u." In Section 2 we derive the "ergodic" definition, i.e. the definition which is possible to interpret by the aid of relative frequencies in a single realisation. This definition has been treated previously by Leadbetter, 1966 [5], and it turns out to be related to Kac and Slepian's horizontal window definition. In Section 3 we give a representation of ξ(t) near a maximum as the difference between a non-stationary normal process and a deterministic process, and in Section 4 we examine these processes as u → -∞. We have then to distinguish between two cases. A: Regular case. r(t) = 1 -λ&lt;sub&gt;2t&lt;/sub&gt;&lt;sup&gt;2&lt;/sup&gt;/2 + λ&lt;sub&gt;4&lt;/sub&gt; t&lt;sup&gt;4&lt;/sup&gt;/4! - λ&lt;sub&gt;6&lt;/sub&gt; t&lt;sup&gt;6&lt;/sup&gt;/6! + o(t&lt;sup&gt;6&lt;/sup&gt;) as t → 0, where the positive λ&lt;sub&gt;2k&lt;/sub&gt; are the spectral moments. Then it is proved that if ξ(t) has a maximum of height u at t = 0 then, as u → -∞, egin{align*} (lambda_2lambda_6 - lambda_4^2)(lambda_4 - lambda_2^2)^{-1}{xi((lambda_2lambda_6 - lambda_4^2)^{-frac{1}{2}}(lambda_4 - lambda_2^2)^{frac{1}{2}}t|u|^{-1}) - u} \ sim |u|^{-3}{t^4/4! + omega(lambda_4 - lambda_2^2)^{frac{1}{2}}lambda_2^ {-frac{1}{2}}t^3/3! - zeta(lambda_4 - lambda_2^2)lambda_2 ^{-1}t^2/2}end{align*} where ω and ζ are independent random variables (rv), ω has a standard normal distribution and ζ has the density &lt;latex&gt;z exp (-z), z &gt; 0&lt;/latex&gt;. Thus, in the neighborhood of a very low maximum the sample functions are fourth degree polynomials with positive t&lt;sup&gt;4&lt;/sup&gt;-term, symmetrically distributed t&lt;sup&gt;3&lt;/sup&gt;-term, and a negatively distributed t&lt;sup&gt;2&lt;/sup&gt;-term but without t-term. B: Irregular case. r(t) = 1 - λ&lt;sub&gt;2t&lt;/sub&gt;&lt;sup&gt;2&lt;/sup&gt;/2 + λ&lt;sub&gt;4t&lt;/sub&gt;&lt;sup&gt;4&lt;/sup&gt;/4! - λ&lt;sub&gt;5&lt;/sub&gt;|t|&lt;sup&gt;5&lt;/sup&gt;/5! + o(t&lt;sup&gt;5&lt;/sup&gt;) as t → 0, where &lt;latex&gt;λ5 &gt; 0&lt;/latex&gt;. Now ξ(tu&lt;sup&gt;-2&lt;/sup&gt;) - u ∼ |u|&lt;sup&gt;-5&lt;/sup&gt;{λ&lt;sub&gt;2&lt;/sub&gt;λ&lt;sub&gt;5&lt;/sub&gt;(λ&lt;sub&gt;4&lt;/sub&gt; - λ&lt;sub&gt;2&lt;/sub&gt;&lt;sup&gt;2&lt;/sup&gt;)&lt;sup&gt;-1&lt;/sup&gt; |t|&lt;sup&gt;3&lt;/sup&gt;/3! + (2λ&lt;sub&gt;5&lt;/sub&gt;)&lt;sup&gt;1/2&lt;/sup&gt; ω(t) - ζ(λ&lt;sub&gt;4&lt;/sub&gt; - λ&lt;sub&gt;2&lt;/sub&gt;&lt;sup&gt;2&lt;/sup&gt;)λ&lt;sub&gt;2&lt;/sub&gt; &lt;sup&gt;-1&lt;/sup&gt;t&lt;sup&gt;2&lt;/sup&gt;/2} where ω(t) is a non-stationary normal process whose second derivative is a Wiener process, independent of ζ which has the density &lt;latex&gt;z exp (-z), z &gt; 0&lt;/latex&gt;. The term λ&lt;sub&gt;5&lt;/sub&gt;|t|&lt;sup&gt;5&lt;/sup&gt;/5! "disturbs" the process in such a way that the order of the distance which can be surveyed is reduced from 1/|u| (in Case A) to 1/|u|&lt;sup&gt;2&lt;/sup&gt;. The results are used in Section 5 to examine the distribution of the wave-length and the crest-to-trough wave-height, i.e., the amplitude, discussed by, among others, Cartwright and Longuet-Higgins, 1956 [1]. One hypothesis, sometimes found in the literature, [10], states that the amplitude has a Rayleigh distribution and is independent of the mean level. According to this hypothesis the amplitude is of the order 1/|u| as u → -∞ while the results of this paper show that it is of the order 1/|u|&lt;sup&gt;3&lt;/sup&gt;.},
  author       = {Lindgren, Georg},
  issn         = {0003-4851},
  language     = {eng},
  number       = {6},
  pages        = {1870--1883},
  publisher    = {Institute of Mathematical Statistics},
  series       = {Annals of Mathematical Statistics},
  title        = {Some properties of a normal process near a local maximum},
  volume       = {41},
  year         = {1970},
}