When a mathematical model of random noise is sought, an important problem is how to unify the deterministic character to describe a time-sequence of random phenomena according to the law of causality and the probabilistic character which accounts for the accidental character existing in actual random phenomena. More concretely, what we must consider are; 1) type of deterministic expression (temporal functions) to be used, 2) kind of probability distribution to be chosen, 3) how to incorporate the probability distribution into the deterministic expression. As mentioned in a previous paper, two models of white noise due to S. O. Rice do not seem to give an organic unity of the deterministic and the probabilistic characters of random noise since the probability distributions are brought into the Fourier series representations at the outset independently of the lapse of time to express a possible variety of amplitude or phase at an arbitrarily fixed time. The truth is that in random processes existing in the physical world all possible varieties of amplitude, phase or other physical quantities really appear in a sufficiently long interval of time. In this paper, we have theoretically introduced a new mathematical model of random noise expressed in terms of uinformly almost periodic functions consisting of arbitrary component waves from the generalized view-point containing the trigonometric series type U. A. P. functions reported in the previous paper. That is, I_N(t)=Σ^^N__<n=1><C_nF(θ_n)>, θ_n≜2π(f_nt+φ_n) (mod 2π) with C_n=C_0(∀n), where F(θ) shows an arbitrary single-valued function under the condition of Eq. (2) and all the frequency ratios (such as f_1/f_2/, f_2/f_3, ……) form a set of irrational numbers. Now it is not necessary to introduce any probability distribution law at the outset into the new model, because the probability distribution is automatically formed in the course of time. Thus, a probability density function P(I_N) or cumulative probability distribution Q(I_N) can be expressed in terms of the statistical Hermite series expansion: P(Y)=n(Y){1+Σ^^∞__<n=2>(-1)^nD_n/σ^<2n>H_<2n>(Y)}=n(Y)+Σ^^∞__<n=2>(-1)^nD_n/σ^<2n>n^<(2n)>(Y), Q(Y)=Φ(Y)+Σ^^∞__<n=2>(-1)^nD_n/σ^<2n>n^<(2n-1)>(Y) with a dimensionless variable Y≜I_N/σ, as the solution of an intergral equation (16) derived through the calculation of characteristic function g(u) (cf. Eqs. (3) and (10)). Here, n(Y)&trie:exp(-Y^2/2)/√<2π>, Φ(Y)&trie:∫^^Y__<-∞>n(Y)dY and σ^2=NC_0^2/2. It should be noted that P(I_N) is asymptotically normal distribution as N tends to infinity and the choice of F(θ) gives a substantial contribution to the speed of convergence tending to normal distribution. Furthermore, the tables (cf. Tables 1 and 2) of explicit expressions P(Y)(or Q(Y)) in the form of statistical Hermite series expansion are given corresponding to several concrete cases where the component wave F(θ_n) of I_N(t) has respectively the specialized forms. Finally, in a special model of random noise formed in terms of trigonometric series consisting of U. A. P. functions, the characteristic that the nth order moments (n=2, 4, 6, 8, ……) of I_N(t) become asymptotically those of normal distribution (with mean zero and variance σ^2) for large N are discussed.
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