Harmonic analysis for a class of multiplicative processes

The harmonic analysis of certain multiplicative processes of the form g(t)X(t) is considered, where g is a deterministic function, and the stochastic process X(t) is of the form X(t)=\sum X_{n}l_{[n \alpha , (n+l) \alpha]}(t) , where a is a positive constant and the X_{n}, n=0, \pm 1,\pm 2, \cdots 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 \alpha , then g(t)X(t) has an autocovariance in the Wiener sense equal to the product of the Wiener autocovariances of its factors, C_{gx} = C_{g}C_{x} . Some important cases are examined where the autocovariance of the multiplicative process exists but cannot be obtained multiplicatively.