Small and Large Scale Behavior of the Poissonized Telecom Process |
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Authors: | Cohen Serge Taqqu Murad S |
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Institution: | (1) Laboratoire de Statistique et Probabilités, UMR C5583, Université Paul Sabatier, 118, Route de Narbonne, 31062 Toulouse Cédex 4, France;(2) Department of Mathematics and Statistics, Boston University, 111 Cummington St., Boston, MA 02215, USA |
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Abstract: | The stable Telecom process has infinite variance and appears as a limit of renormalized renewal reward processes. We study its Poissonized version where the infinite variance stable measure is replaced by a Poisson point measure. We show that this Poissonized version converges to the stable Telecom process at small scales and to the Gaussian fractional Brownian motion at large scales. This process is therefore locally as well as asymptotically self-similar. The value of the self-similarity parameter at large scales, namely the self-similarity parameter of the limit fractional Brownian motion, depends on the form the Poissonized Telecom process. The Poissonized Telecom process is a Poissonized mixed moving average. We investigate more general Poissonized mixed moving averages as well. |
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Keywords: | fractional Brownian motion self-similarity local self-similarity Telecom process |
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