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A mixed Gaussian fractional process $\{Y(t)\}_{t \in {\Bbb R}} = \{PX(t)\}_{t\in {\Bbb R}}$ is a multivariate stochastic process obtained by pre-multiplyinga vector of independent, Gaussian fractional process entries $X$ by anonsingular matrix $P$. It is interpreted that $Y$ is observable, while $X$ isa hidden process occurring in an (unknown) system of coordinates $P$. Mixedprocesses naturally arise as approximations to solutions of physically relevantclasses of multivariate fractional SDEs under aggregation. We propose asemiparametric two-step wavelet-based method for estimating both the demixingmatrix $P^{-1}$ and the memory parameters of $X$. The asymptotic normality ofthe estimators is established both in continuous and discrete time. Monte Carloexperiments show that the finite sample estimation performance is comparable tothat of parametric methods, while being very computationally efficient. Asapplications, we model a bivariate time series of annual tree ring widthmeasurements, and establish the asymptotic normality of the eigenstructure ofsample wavelet matrices.

Two-step wavelet-based estimation for mixed Gaussian fractional processes