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In this paper, a unified theory of the statistical treatment for the output probability distribution is newly proposed by introducing a statistical Lagrange series expansion method. In the development of the theory, the assumed situation is that a general random process of arbitrary distribution type is passed through a non-stationary stochastic system of memory type with an arbitrary non-linear feedback element. For the purpose of finding the effect of an arbitrary non-linear feedback element on the output probability distribution, the explicit expressions of the probability distributions are derived in general forms of non-orthogonal expansion series, reflecting the various effects of the forward linear element of the system into the first term. Further, in view of the arbitrariness of the input characteristics, the possible variety of non-linear element and fluctuation forms of system parameters, and the complexity of the statistical treatment involved, the validity of theoretical expressions is experimentally confirmed by the method of digital simulation.

Multi-Variate Probability Expressions for Non-Stationary Response of a Stochastic System with a Non-Linear Feedback Element and Random Input