Abstract :
This paper compares models of fractional processes and associated weak convergence
results based on moving average representations in the time domain with
spectral representations+ Both approaches have been applied in the literature on
fractional processes+ We point out that the conventional forms of these models
are not equivalent, as is commonly assumed, even under a Gaussianity assumption+
We show that it is necessary to distinguish between “two-sided” processes
depending on both leads and lags from one-sided or “causal” processes, because
in the case of fractional processes these models yield different limiting properties+
We derive new representations of fractional Brownian motion and show how
different results are obtained for, in particular, the distribution of stochastic integrals
in the multivariate context+ Our results have implications for valid statistical
inference in fractional integration and cointegration models+
