Harmonic analysis for a class of multiplicative processes

The harmonic analysis of certain multiplicative processes of the form is considered, where is a deterministic function, and the stochastic process is of the form , where a is a positive constant and the are independent and identically distributed random variables with zero means and finite variances. In particular, we show that if g is Riemann integrable and periodic, with period incommensurate with , then has an autocovariance in the Wiener sense equal to the product of the Wiener autocovariances of its factors, . Some important cases are examined where the autocovariance of the multiplicative process exists but cannot be obtained multiplicatively.