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Limit theorems for certain functionals associated to stable processes in a Hölder space. (Théorèmes limites pour certaines fonctionnelles associées aux processus stables sur l’espace de Hölder.) (French) Zbl 0995.60037
Let \(\{L_t^x : t\geq 0, x\in\mathbb R\}\) be the local time of an \(\alpha\)-Levy motion \((Z_t)_{t\geq 0}\) where \(1<\alpha\leq 2\). The aim of the present paper is to investigate \(L_t^x\) as function of the two variables \(t\) and \(x\) jointly. For example, a Hölder type estimate is proved for \((t,x)\to L_t^x\). Furthermore, the authors verify Hölder conditions for certain fractional derivatives of \(L_t^x\) (here the derivative is taken with respect to \(x\in\mathbb R\)). As a consequence, limits of some special functionals of the motion \((Z_t)_{t\geq 0}\) may be described by means of fractional derivatives of \(L_t^x\).

60F25 \(L^p\)-limit theorems
60G52 Stable stochastic processes
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