Hausdorff dimension of Weierstrass-Mandelbrot process
type: Item
instance of: scholarly article
title: Hausdorff dimension of Weierstrass-Mandelbrot process
zbMATH Open document ID: 1005.60086
DOI: 10.1016/S0167-7152(01)00195-X
author: Jerzy Szulga
published in: Statistics \& Probability Letters
publication date: 2002-09-05T00:00:00Z
review text: Let \(W_r(t)= \sqrt{(\ln r)}\sum _{n=-\infty} ^{n=\infty} (1-\exp(ir^nt)) r^{-nH} \eta_n\), where \(\eta_n= \exp(i\theta_n)\), \( \theta_n\) are i.i.d. uniform variables on \([0,2\pi]\), \(H\in (0,1)\) and \(r\in (1,\infty)\). Such process is usually called Weierstrass-Mandelbrot process and as \(r\to 1\), \(W_r(t)\) converges in distribution to a complex fractional Brownian motion with the Hurst parameter equal to \(H\). The author proves that Weierstrass-Mandelbrot process and fractional Brownian motion have the same Hausdorff dimension of path and ranges, and, thus, verify the Berry-Lewis hypothesis [\textit{M. V. Berry} and \textit{Z. V. Lewis}, Proc. R. Soc. Lond., Ser. A 370, 459-484 (1980; Zbl 0435.28008)].
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60J99
Mathematics Subject Classification ID: 60G17
zbMATH DE Number: 1796691
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: fractal dimension
zbMATH Keywords: fractional Brownian motion
zbMATH Keywords: Weierstrass-Mandelbrot process
zbMATH Keywords: small ball probability
zbMATH Keywords: law tails
zbMATH Keywords: characteristic function
MaRDI profile type: Publication
cites work: On the Weierstrass-Mandelbrot fractal function
cites work: Q3671491
cites work: The Hausdorff dimension of graphs of Weierstrass functions
cites work: Q4721321
cites work: Q4154814
cites work: Convergence of the Weierstrass-Mandelbrot process to fractional Brownian motion
cites work: Q4266577
cites work: The Weierstrass-Mandelbrot process revisited
description: scientific article; zbMATH DE number 1796691