Some Properties of a Normal Process Near a Local Maximum

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 -λ₂t²/2 + λ₄ t⁴/4! - λ₆ t⁶/6! + o(t⁶) as t → 0, where the positive λ₂ₖ are the spectral moments. Then it is proved that if ξ(t) has a maximum of height u at $t = 0$ then, as u → -∞, {align*} (λ_2λ_6 - λ_4^2)(λ_4 - λ_2^2)⁻¹\{ξ((λ_2λ_6 - λ_4^2)-1/2(λ_4 - λ_2^2)1/2t|u|⁻¹) - u\} \\ ~ |u|⁻³\{t^4/4! + ω(λ_4 - λ_2^2)1/2λ_2^ {-1/2}t^3/3! - ζ(λ_4 - λ_2^2)λ_2 ⁻¹t^2/2\}{align*} where ω and ζ are independent random variables (rv), ω has a standard normal distribution and ζ 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 t⁴-term, symmetrically distributed t³-term, and a negatively distributed t²-term but without t-term. B: Irregular case. r(t) = 1 - λ₂t²/2 + λ₄t⁴/4! - λ₅|t|⁵/5! + o(t⁵) as t → 0, where λ₅ > 0. Now ξ(tu⁻²) - u ~ |u|⁻⁵\λ₂λ₅(λ₄ - λ₂²)⁻¹ |t|³/3! + (2λ₅)1/2 ω(t) - ζ(λ₄ - λ₂²)λ₂ ⁻¹t²/2\ where ω(t) is a non-stationary normal process whose second derivative is a Wiener process, independent of ζ which has the density z exp (-z), z > 0. The term λ₅|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 → -∞ while the results of this paper show that it is of the order 1/|u|³.