## Bernoulli

• Bernoulli
• Volume 5, Number 2 (1999), 191-207.

### Estimation of Rényi exponents in random cascades

#### Abstract

We consider statistical estimation of the Rényi exponent $τ (h)$, which characterizes the scaling behaviour of a singular measure $μ$ defined on a subset of $R d$. The Rényi exponent is defined to be $lim δ →0 [{logM δ (h)}/(-logδ)]$, assuming that this limit exists, where $M δ (h)=∑ i μ h(Δ i)$ and, for $δ >0$, ${ Δ i}$ are the cubes of a $δ$-coordinate mesh that intersect the support of $μ$. In particular, we demonstrate asymptotic normality of the least-squares estimator of $τ (h)$ when the measure $μ$ 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.

#### Article information

Source
Bernoulli, Volume 5, Number 2 (1999), 191-207.

Dates
First available in Project Euclid: 5 March 2007

https://projecteuclid.org/euclid.bj/1173147902

Mathematical Reviews number (MathSciNet)
MR1681694

Zentralblatt MATH identifier
0953.62088

#### Citation

Troutman, Brent M.; Vecchia, Aldo V. Estimation of Rényi exponents in random cascades. Bernoulli 5 (1999), no. 2, 191--207. https://projecteuclid.org/euclid.bj/1173147902

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