  The linear fractional stable motion generalizes two prominent classes of
stochastic processes, namely stable L\'evy processes, and fractional Brownian
motion. For this reason it may be regarded as a basic building block for
continuous time models. We study a stylized model consisting of a superposition
of independent linear fractional stable motions and our focus is on parameter
estimation of the model. Applying an estimating equations approach, we
construct estimators for the whole set of parameters and derive their
asymptotic normality in a high-frequency regime. The conditions for consistency
turn out to be sharp for two prominent special cases: (i) for L\'evy processes,
i.e. for the estimation of the successive Blumenthal-Getoor indices, and (ii)
for the mixed fractional Brownian motion introduced by Cheridito. In the
remaining cases, our results reveal a delicate interplay between the Hurst
parameters and the indices of stability. Our asymptotic theory is based on new
limit theorems for multiscale moving average processes.
