The authors give a deep and detailed survey on generalizations of fractional Brownian motion, with main interest in time-dependent self-similarity caused by time-dependent self-similarity parameter (Hurst exponent). Among other things they treat fARIMA (fractional autoregressive integrated moving average) and gfARIMA (generalizations of the latter) processes and discuss the topic of fractal, Hausdorff and local dimensions of sample paths. For generalized fractional Brownian motion they state a law of the iterated logarithm. Then they turn their attention to the problem of estimating the scaling function (the time-dependent similarity parameter). For this they propose the use of wavelet analysis and develop a discrete numerical scheme for its implementation. By several simulations and illuminating case studies (some artificial, some taken from diverse real applications) and graphically presented results they demonstrate the practicability of their methods.
For the entire collection see [Zbl 1009.00011].