Selected topics in the generalized mixed set-indexed fractional Brownian motion
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.
scientific article; zbMATH DE number 7374938
scientific article