Upper and lower classes for \(\mathbb{L}^ 2\)- and \(\mathbb{L}^ p\)-norms of Brownian motion and norms of \(\alpha\)-stable motion
Let \(X (\theta, t) = \int^t_{\theta t} |B (s) |^pds\), \(p \geq 1\), \(t \geq 0\), \(\theta \in [0,1)\), where \(B\) is standard Brownian motion. Integral tests for upper and lower functions for \(X (\theta,t)\) are given in the case when \(p = 2\). Next, the growth of \(X (\theta,t)\) is studied for any \(p \geq 1\). Finally, the growth of \(|\xi (\cdot) I_{[\theta t,t)} (\cdot) |\) is studied. Here \(|\cdot |\) is a measurable norm on \(\{(f : {\mathbf R}^+ \to {\mathbf R}) : f\) right-continuous with compact support\} satisfying some regularity assumptions, and \(\xi\) is a right-continuous \(\alpha\)-stable motion with \(\alpha \in (0,1) \cup (1,2)\) and skewness \(\beta \in [- 1,1]\), or with \((\alpha, \beta) = (1,0)\).
scientific article; zbMATH DE number 803379
scientific article