A stochastic study of the non-stationary non-negative random process based on two models of time series: multiplication and addition

As is well-known, there are two contrasting and fundamental models for simulating actual random processes with time series: a multiplication model and an addition model. In this letter, two explicit expressions of the probability density function for a non-stationary non-negative random process (a statistical Laguerre expansion type and a statistical Hermite expansion type) are derived from the above two fundamental viewpoints of modeling a time series, in relation to the statistical method described in a previous paper by the authors, in which the analysis was based on the use of a Hankel transform type characteristic function. The unified theory introduced in this previous paper can be obtained by a very simplified calculation as compared with that of the previous study, by the natural introduction of a random time series multiplication model.