On the increments of some extensions of the fractional Brownian motion

Let {B-H(t), t >= 0} be a fractional Brownian motion with Hurst index 0 < H < 1. Given that the process B-H has stationary increments, a detailed understanding of the characteristics of small, medium, and large increments is essential. This exploration extends to the QHASI class presented as a collection of centered Gaussian processes. This class is mainly characterized by a self-similarity, a quasi-helix and an approximately stationary increments assumptions. However, the analysis of these increments requires a different approach when considering the new extension of B-H proposed in some previous research. This investigation reveals significant differences in the behavior of increments within this extended framework.