Estimation of Rényi exponents in random cascades

Abstract

We consider statistical estimation of the Rényi exponent $\tau \left(h\right)$, which characterizes the scaling behaviour of a singular measure $\mu$ defined on a subset of $R^d$. The Rényi exponent is defined to be ${lim}_{\delta \to 0}\left[\right\{log{M}_{\delta }\left(h\right)\}/(-log\delta \left)\right]$, assuming that this limit exists, where $M_{\delta }\left(h\right)={\sum }_i{\mu }^h\left({\Delta }_i\right)$ and, for $\delta >0$, $\left\{{\Delta }_i\right\}$ are the cubes of a $\delta$-coordinate mesh that intersect the support of $\mu$. In particular, we demonstrate asymptotic normality of the least-squares estimator of $\tau \left(h\right)$ when the measure $\mu$ is generated by a particular class of multiplicative random cascades, a result which allows construction of interval estimates and application of hypothesis tests for this scaling exponent. Simulation results illustrating this asymptotic normality are presented.