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On some possible generalizations of fractional Brownian motion. (English) Zbl 1068.82518
Summary: Fractional Brownian motion (fBm) can be generalized to multifractional Brownian motion (mBm) if the Hurst exponent $H$ is replaced by a deterministic function $H\left(t\right)$. The two possible generalizations of mBm based on the moving average representation and the harmonizable representation are first shown to be equivalent up to a multiplicative deterministic function of time by S. Cohen (1999) using the Fourier transform method. In this letter, the authors give an alternative verification of such an equivalence based on the direct computation of the covariances of these two Gaussian processes. There also exists another equivalent representation of mBm, which is a variant version of the harmonizable representation. Finally, they consider a generalization based on the Riemann-Liouville fractional integral, and study the large time asymptotic properties of this version of mBm.
##### MSC:
 82B41 Random walks, random surfaces, lattice animals, etc. (statistical mechanics) 60K40 Physical applications of random processes