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Local nondeterminism and local times for stable processes. (English) Zbl 0659.60106

Our main theorem gives sufficient conditions for symmetric stable processes and fields to have a jointly continuous local time. The approach is through the ${L}^{p}$-representation for such processes. We develop a measure of dependence for vectors in a normed linear space and use that to analyze the probabilistic independence of the increments of a stable process. Local nondeterminism is defined for stable processes and shown to be equivalent to “locally approximately independent increments.” Sufficient conditions for several classes of stable processes to be local nondeterministic are given.

These ideas are extended to multidimensional stable random fields and we prove existence of jointly continuous local times. The results extend most Gaussian results to their stable analogs.

Reviewer: J.P.Nolan

##### MSC:
 60J55 Local time, additive functionals 60G17 Sample path properties 60G60 Random fields
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