A trial of a new formation of the random noise model by use of arbitrary uniformly almost periodic functions

In this paper, a new mathematical model of random noise in terms of the uniformly almost periodic functions consisting of arbitrary component waveforms is theoretically introduced from a generalized viewpoint containing the uniformly almost periodic functions of cosine wave type reported in the previous paper. That is, IN(t) = Σn=1N CnF(θn), θn = 2π × (fnt + n) (mod 2π) with Cn = C0 (∀n), where F(θ) shows an arbitrary single-valued function under a certain condition and all the frequency ratios (such as f1/f2, f2/f3,…) form a set of irrational numbers. Hereupon, the explicit expressions of the probability density function, PN(I), in the form of the statistical Hermite series expansion, are given corresponding to several concrete cases where the component wave F(θ) has the special form. Further, the characteristics that PN(I) is asymptotically a Gaussian distribution as N tends to infinity, and the fact that the representative pattern of the component waveforms and the choice of shape factors characterizing the specific F(θ) give a substantial contribution to the speed of convergence tending to the Gaussian distribution, are investigated. The δ-correlation property of the new random noise model, which is another feature of random noise, is also discussed. In view of the variety of component waveforms, the changeability of the number of component waves included, and the arbitrariness of irrational frequency ratios of component waves, the following properties are quantitatively considered, especially from an experimental viewpoint: (1) the non-Gaussian property at a finite value of N and the asymptotic property to a Gaussian distribution at a large value of N; (2) the effects of the representative pattern of F(θ) of the component waves and the choice of shape factors characterizing the specific F(θ) on the speed of convergence to the Gaussian properties, from two points of the distribution form and the statistical moments; and (3) the correlation property as N becomes large