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Harmonic analysis for a class of multiplicative processes

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2 Author(s)

The harmonic analysis of certain multiplicative processes of the formg(t)X(t)is considered, wheregis a deterministic function, and the stochastic processX(t)is of the formX(t)=sum X_{n}l_{[n alpha , (n+l) alpha]}(t), where a is a positive constant and theX_{n}, n=0, pm 1,pm 2, cdotsare 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 withalpha, theng(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.

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Information Theory, IEEE Transactions on  (Volume:33 ,  Issue: 1 )