## The Annals of Statistics

- Ann. Statist.
- Volume 32, Number 3 (2004), 1222-1260.

### Estimation of fractal dimension for a class of non-Gaussian stationary processes and fields

Grace Chan and Andrew T. A. Wood

#### Abstract

We present the asymptotic distribution theory for a class of increment-based estimators of the fractal dimension of a random field of the form *g*{*X*(*t*)}, where *g*:**R**→**R** is an unknown smooth function and *X*(*t*) is a real-valued stationary Gaussian field on **R**^{d}, *d*=1 or 2, whose covariance function obeys a power law at the origin. The relevant theoretical framework here is “fixed domain” (or “infill”) asymptotics. Surprisingly, the limit theory in this non-Gaussian case is somewhat richer than in the Gaussian case (the latter is recovered when *g* is affine), in part because estimators of the type considered may have an asymptotic variance which is random in the limit. Broadly, when *g* is smooth and nonaffine, three types of limit distributions can arise, types (i), (ii) and (iii), say. Each type can be represented as a random integral. More specifically, type (i) can be represented as the integral of a certain random function with respect to Lebesgue measure; type (ii) can be represented as the integral of a second random function with respect to an independent Gaussian random measure; and type (iii) can be represented as a Wiener–Itô integral of order 2. Which type occurs depends on a combination of the following factors: the roughness of *X*(*t*), whether *d*=1 or *d*=2 and the order of the increment which is used. Another notable feature of our results is that, even though the estimators we consider are based on a variogram, no moment conditions are required on the observed field *g*{*X*(*t*)} for the limit theory to hold. The results of a numerical study are also presented.

#### Article information

**Source**

Ann. Statist. Volume 32, Number 3 (2004), 1222-1260.

**Dates**

First available in Project Euclid: 24 May 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1085408501

**Digital Object Identifier**

doi:10.1214/009053604000000346

**Mathematical Reviews number (MathSciNet)**

MR2065204

**Zentralblatt MATH identifier**

1046.62103

**Subjects**

Primary: 62M99: None of the above, but in this section

Secondary: 62E20: Asymptotic distribution theory

**Keywords**

Gaussian random measure Hermite polynomial increments stationary random field variogram Wiener–Itô integral

#### Citation

Chan, Grace; Wood, Andrew T. A. Estimation of fractal dimension for a class of non-Gaussian stationary processes and fields. Ann. Statist. 32 (2004), no. 3, 1222--1260. doi:10.1214/009053604000000346. https://projecteuclid.org/euclid.aos/1085408501.