Formal definition for Hurst exponent using self-similar stochastic processes.

Question

I know more or less what Hurst exponent implies and the original definition based on the re-escaled range, but for my bachelor thesis I need to define it from a formal definition using self-similar stochastic processes. I have been told to search for it in the book "Long-Memory Processes" but I have only found the original one. Could you please give me some hints or reccomend where I could look for that kind of definition?

Answer 1

In case someone was wondering, it can be definined at the same time self-similar process is defined as:

A random process X is H−self-similar (also self-similar of parameter H) if there exists $H ≥ 0$ such that $X_{at} \sim a^H X_t$ for all $a > 0$ and each $t ≥ 0$, where $X_t\sim Y_t$ denotes that the finit joint distribution of $X_t$ coincides with the one of $Y_t$.