Some Properties of a Normal Process Near a Local Maximum

Abstract

Consider a stationary normal process $\xi(t)$ 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 $u \rightarrow -\infty$, 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 $\xi(t)$ 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 $u \rightarrow -\infty$. We have then to distinguish between two cases. A: Regular case. $r(t) = 1 -\lambda_2t^2/2 + \lambda_4 t^4/4! - \lambda_6 t^6/6! + o(t^6)$ as $t \rightarrow 0$, where the positive $\lambda_{2k}$ are the spectral moments. Then it is proved that if $\xi(t)$ has a maximum of height $u$ at $t = 0$ then, as $u \rightarrow -\infty$, \begin{align*} (\lambda_2\lambda_6 - \lambda_4^2)(\lambda_4 - \lambda_2^2)^{-1}\{\xi((\lambda_2\lambda_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 $\omega$ and $\zeta$ are independent random variables (rv), $\omega$ has a standard normal distribution and $\zeta$ has the density $z \exp (-z), z > 0$. Thus, in the neighborhood of a very low maximum the sample functions are fourth degree polynomials with positive $t^4$-term, symmetrically distributed $t^3$-term, and a negatively distributed $t^2$-term but without $t$-term. B: Irregular case. $r(t) = 1 - \lambda_2t^2/2 + \lambda_4t^4/4! - \lambda_5|t|^5/5! + o(t^5)$ as $t \rightarrow 0$, where $\lambda_5 > 0$. Now $\xi(tu^{-2}) - u \sim |u|^{-5}\{\lambda_2\lambda_5(\lambda_4 - \lambda_2^2)^{-1} |t|^3/3! + (2\lambda_5)^{\frac{1}{2}} \omega(t) - \zeta(\lambda_4 - \lambda_2^2)\lambda_2 ^{-1}t^2/2\}$ where $\omega(t)$ is a non-stationary normal process whose second derivative is a Wiener process, independent of $\zeta$ which has the density $z \exp (-z), z > 0$. The term $\lambda_5|t|^5/5!$ "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 $1/|u|^2$. 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 $u \rightarrow -\infty$ while the results of this paper show that it is of the order $1/|u|^3$.