Estimation of fractal dimension for a class of non-Gaussian stationary processes and fields
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一类非高斯平稳过程与随机场的分形维数估计
- 研究问题
- 对平稳高斯随机场经过未知光滑非线性变换形成的非高斯场，基于增量的分形维数估计量具有怎样的固定域极限分布？
- 方法与思路
- 作者在固定域加密抽样框架下，依据场的粗糙度、空间维度与增量阶数分析极限分布，并给出随机积分表示及数值研究。
- 材料陈述的贡献
- 摘要称可出现三类极限分布及随机极限方差，且所研究估计量的极限理论不要求观测场满足矩条件。
- 阅读关注点
- 应核对协方差原点幂律与变换光滑性条件、三类极限的分界、随机方差如何影响推断，以及增量阶数的选择依据。
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原文摘要
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 Rd, 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.