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A Poisson bridge between fractional Brownian motion and stable Lévy motion. (English) Zbl 1087.60080
Summary: We study a non-Gaussian and non-stable process arising as the limit of sums of rescaled renewal processes under the condition of intermediate growth. The process has been characterized earlier by the cumulant generating function of its finite-dimensional distributions. Here, we derive a more tractable representation for it as a stochastic integral of a deterministic function with respect to a compensated Poisson random measure. Employing the representation we show that the process is locally and globally asymptotically self-similar with fractional Brownian motion and stable Lévy motion as its tangent limits.

MSC:
60K99 Special processes
60G57 Random measures
90B18 Communication networks in operations research
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