Bilateral Canonical Cascades: Multiplicative Refinement Paths to Wiener's and Variant Fractional Brownian Limits

Abstract

The original density is 1 for $t\in (0,1)$, $b$ is an integer base ($b\geq 2$%), and $p\in (0,1)$ is a parameter. The first construction stage divides the unit interval into $b$ subintervals and multiplies the density in each subinterval by either 1 or -1 with the respective frequencies of $\frac{1% }{2}+\frac{p}{2}$ and ${1/2}-\frac{p}{2}$. It is shown that the resulting density can be renormalized so that, as $n\to \infty $ ($n$ being the number of iterations) the signed measure converges in some sense to a non-degenerate limit. If $H=1+\log_{b}$ $p>{1}/{2}$, hence $p>b^{{-1}/{% 2}}$, renormalization creates a martingale, the convergence is strong, and the limit shares the H\"{o}lder and Hausdorff properties of the fractional Brownian motion of exponent $H$. If $H\leq {1}/{2}$, hence $p\leq b^{{-1}/{2}%}$, this martingale does not converge. However, a different normalization can be applied, for $H\leq {1/2}$ to the martingale itself and for $H>% {1/2}$ to the discrepancy between the limit and a finite approximation. In all cases the resulting process is found to converge weakly to the Wiener Brownian motion, independently of $H$ and of $b$. Thus, to the usual additive paths toward Wiener measure, this procedure adds an infinity of multiplicative paths.