A random variable $X(t)$ is said to be monoscaling if $$ X(t) = a^{-H}X(at).$$ $H$ is called the Hurst exponent, and $a$ is a scaling factor. A key model of statistical monoscaling is the fractional Brownian motion.

More generally, a variable can be multiscaling, meaning each moment changes to a different degree upon scaling $t$ to $at$.

How does one model statistical multiscaling?

  • $\begingroup$ The terms monoscaling and multiscaling are new to me. By multi-scaling, do you mean that the variance, skewness, and kurtsosis also have the sort of fractionally autoregressive properties? $\endgroup$
    – John
    Oct 11, 2019 at 17:40
  • $\begingroup$ Yes, exactly. The $q$th order moments at aggregation scale $a$ are $M(q,a) = \langle X(at)^q \rangle$. If $M(q,a) \propto a^{qH}$, the random variable $X(t)$ is said to be monoscaling, while if $M(q,a) \propto a^{\tau(q)}$, where $\tau(q)$ is some non-linear function, $X$ is said to be multiscaling. "multi" denotes the need for multiple Hurst exponents to describe the scale dependence of the random variable, if I understand properly. I have some experimental data that exhibit multiscaling and I don't know how to model them. $\endgroup$ Oct 11, 2019 at 18:53
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    $\begingroup$ I can't say that I can help much. You might try starting with Mandelbrot et al's "A Multifractal Model of Asset Returns." (Cowles #1164). $\endgroup$
    – John
    Oct 11, 2019 at 20:09


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