Fractional multiplicative processes

TL;DR

This paper extends multiplicative cascade models by allowing negative weights, resulting in martingales that converge to self-similar functions with properties akin to fractional Brownian motion, revealing new stable laws and divergence behaviors.

Contribution

It introduces a novel class of self-similar processes via negative weights in multiplicative cascades, linking them to fractional Brownian motion and stable laws.

Findings

For H in (1/2,1), the process converges to a self-similar function with fractional Brownian motion properties.

For H in (0,1/2], the process diverges but normalized versions converge to Brownian motion.

New stable law type emerges, stable under random weighted averaging, satisfying a functional equation.

Abstract

Statistically self-similar measures on $[0,1]$ are limit of multiplicative cascades of random weights distributed on the $b$-adic subintervals of $[0,1]$. These weights are i.i.d, positive, and of expectation $1/b$. We extend these cascades naturally by allowing the random weights to take negative values. This yields martingales taking values in the space of continuous functions on $[0,1]$. Specifically, we consider for each $H\in (0,1)$ the martingale $(B_{n})_{n\geq1}$ obtained when the weights take the values $-b^{-H}$ and $b^{-H}$, in order to get $B_n$ converging almost surely uniformly to a statistically self-similar function $B$ whose H\"{o}lder regularity and fractal properties are comparable with that of the fractional Brownian motion of exponent $H$. This indeed holds when $H\in(1/2,1)$. Also the construction introduces a new kind of law, one that it is stable under random weighted averaging and satisfies the same functional equation as the standard symmetric stable law of index $1/H$. When $H\in(0,1/2]$, to the contrary, $B_n$ diverges almost surely. However, a natural normalization factor $ a_n$ makes the normalized correlated random walk $ B_n / a_n$ converge in law, as $n$ tends to $\infty$, to the restriction to $[0,1]$ of the standard Brownian motion. Limit theorems are also associated with the case $H>1/2$.