The Effect of General Correlations of High Degree on the Expected Number of Level Crossing in the Arbitrary Random Waves Fluctuating only in Positive Region

Abstract

A joint probability density distribution of the instantaneous value X and the slope Y in the arbitrary random wave fluctuating only in positive region can be expressed in the form of a mixed orthonormal expansion series in the statistical Laguerre series and the statistical Hermite series as follows:
P(X, Y)=X^m-1/Γ(m)S^me^-X/S·1/√2πλe^{-(Y-μY)2/2λ}{1+∞Σn=1∞ΣK=1β(n, K)L_n^(m-1)(X/S)H_K(Y-μ_Y/√λ)}.
It must be noticed that the general correlation of high degree between level X and slope Y is reflected in each expansion coefficient β(n, k) (n≠0, k≠0).
While, the expected number M(X) of X per second passing through a level X with positive and negative slopes is given by M(X)=∫∞-∞|Y|P(X, Y)dY.
Therefore, by using an expression of the expected number in the normalized form: M₀(X)=M(x)/(∫∞0M(X)dX), the probability density distribution function of crossing at level X with positive and negative slopes can be expressed as M₀(Z)=P_r(Z;m)+AmP_r⁽¹⁾(Z;m+1)+Bm(m+1)/2!P_r⁽²⁾(Z;m+2)+Cm(m+1)(m+2)/3!P_r⁽³⁾Z;m+3)+Dm(m+1)(m+2)(m+3)/4!P_r⁽⁴⁾Z;m+4……, where Z=X/S, S=μ_x/m, m=μ_x²/δ_x² (μ_x: mean, δ_x²: variance) and P_r(Z;m)≡Z^m-1/Γ(m)e^-Z is a gamma probability density distribution function. In the same way, we can obtain an expression of the probability density distribution function N₀(Z) of crossing at level X with a positive slope in the similar expansion form.
Finally, the detailed experimental considerations to corroborate the above theories are given in the following two cases:
(a) relation between the cumulative probability distribution ∫Z0P(Z)dZ of the instantaneous value and the cumulative probability distribution ∫Z0M₀(Z)dZ(or∫Z0N₀(Z)dZ) of crossing, (b) high correlations β(n, k) between level X and slope Y, by taking the street noise measured in the city of Fukui as an example of the above random wave fluctuating only in positive region.