Selected topics in the generalized mixed set-indexed fractional Brownian motion
type: Item
instance of: scholarly article
title: Selected topics in the generalized mixed set-indexed fractional Brownian motion
zbMATH Open document ID: 1496.60032
author: Arthur Yosef
published in: Journal of Theoretical Probability
publication date: 2021-07-26T00:00:00Z
review text: Let \(\mathcal A\) be a compact set collection on a topological space \((T,\tau).\) A set-indexed fractional Brownian motion (sifbm) of Hurst parameter \(H\in (0, \frac{1}{2}]\) is a centered Gaussian process \(B^H=\{B_A^H:A \in \mathcal A\}\) defined on a probability space \((\Omega, \mathcal F,\mu)\) with the covariance function \[E[B_A^HB_B^H]= \frac{1}{2}[\mu(A)^{2H}+\mu(B)^{2H}-\mu(A\Delta B)^{2H}]\] for all \(A,B\in \mathcal A.\) The author studies properties of a generalized set-indexed fractional Brownian motion (gsifBm). The gsifBm of parameters \(H=\{H_1,\dots,H_n\}, \alpha= \{\alpha_1,\dots,\alpha_n\}, (0,H_i\leq \frac{1}{2}\) and \(\{ \alpha_i \}\) are real not all zero) is the process \[M_A^{H,\alpha}=\sum_{i=1}^n\alpha_iB_A^{H_i}\] for all \(A\in \mathcal A,\) where \(B^{H_i}\) are independent sifBm of parameter \(H_i, 1\leq i \leq n.\) This is a generalized mixed fractional Brownian motion in the set-indexed framework. The author investigates the properties of self-similarity, long-range dependence, Holder continuity, differentiability and Hausdorff dimension for such processes.
Mathematics Subject Classification ID: 60G22
zbMATH DE Number: 7374938
zbMATH Keywords: fractional Brownian motion
zbMATH Keywords: set-indexed fractional Brownian motion
reviewed by: B. L. S. Prakasa Rao
MaRDI profile type: Publication
full work available at URL: s10959-021-01077-6
OpenAlex ID: W3134300943
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DOI: 10.1007/S10959-021-01077-6
description: scientific article; zbMATH DE number 7374938