Stable processes with sample paths in Orlicz spaces
Let \(T\) be a separable metric space, \(\mu\) a \(\sigma\)-finite measure on the Borel sets of \(T\), and \(X\) measurable symmetric \(\alpha\)-stable (S\(\alpha\)S) with the integral representation \[ (X(t); t\in T) =^{(d)} \biggl(\int_E h(t,x)M(dx); t\in T\biggr), \] where \(M\) is an S\(\alpha\)S random measure on \((E,{\mathcal E})\), \(\sigma_\alpha (t)^\alpha = \int_E |h(t,x)|^\alpha m(dx) < \infty\), \(m\) being a \(\sigma\)-finite measure on \((E,{\mathcal E})\), and \(0 < \alpha < 2\). Let \(\psi : R_+ \to R_+\), such that \(\psi(0) = 0\), \(\psi\) is non-decreasing, not bounded, continuous and \(\psi(2u) \leq A\psi(u)\), \(\forall u \leq 0\). The authors prove that \(X\) belongs to the Orlicz space \(L_\psi(T,\mu)\) (i.e. \(\int_T \psi(|X(t)|) \mu(dt) < \infty\) a.s.) if and only if: \[ \int_E|h(.,x)|^\alpha_\psi m(dx) < \infty,\tag{i} \] \[ \int_T \psi(\sigma_\alpha(t)) \mu(dt) < \infty,\tag{ii} \] \[ \int_T \int_E |h(t,x) |^\alpha m(dx)\mu (dt) \int^{\gamma(t,x)}_{c_0 \sigma_\alpha(t)} \psi(r) {dr\over r^{1+\alpha}} < \infty,\tag{iii} \] where \(\gamma(t,x) = |h(t,x)|/ |h(.,x)|_\psi\), and \(|f|_\psi = \inf\{u > 0; \int_T \psi(|f|/u)d\mu \leq 1\}\) and \(c_0 > 0\) is a constant. With some additional assumptions on \(\psi\), the previous equivalence can be reduced to (i) or (ii). An application to the convergence of \(\sum^\infty_1 \xi_i f_i\) in \(L_\psi(T,\mu)\) is also given, \(\xi_i\) (resp. \(f_i\)) being a one-dimensional \(\alpha\)-stable r.v. (resp. a function).
scientific article; zbMATH DE number 828371