On the fractal nature of increments of \(l_p\)-valued Gaussian processes
The author studies the set of points where the oscillation of \(l_p\)-valued Gaussian processes occur infinitely often. Namely, let \(Y(t)=(X_k(t))\) be a sequence of independent Gaussian processes with \(EX_k(t)=0\) and let \(\sigma ^2_k(h)=E(X_k( t+h) -X_k(t))^2\). Further, put \[ \widetilde {\sigma }(p,h)=\begin{cases} \bigl (\sum _{k=1}^\infty \sigma ^s_k(h)\bigr)^{1/s},\;s= \frac {2p}{2-p}, & \text{if} 1\leq p<2;\\ \max _k \sigma _k(h), & \text{if} p\geq 2. \end{cases} \] Then it is shown that under suitable conditions the set \[ \Biggl\{ t\in [0,1];\;\limsup _{h\to 0} \frac {\|Y(t+h)-Y(t)\|_{l_p}}{\widetilde {\sigma }(p,h)(2\log h^{-1})^{1/2}} \geq \alpha \Biggr\}, \quad \alpha \in [0,1], \] is random fractal with the Hausdorff dimension \(1-\alpha ^2\) almost surely. As a consequence the author obtains the estimate of Hausdorff dimensions in the following special cases: \(Y(t)=(c_kX_k(y))\), where \(X_k(t)\) are independent copies of a centered Gaussian process, \((c_k)\in l_p\); \(Y(t)\) is a sequence of independent Ornstein-Uhlenbeck processes.
scientific article; zbMATH DE number 1409136