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Limiting fractal random processes in heavy-tailed systems. (English) Zbl 1186.60018
Lévy-Véhel, Jacques (ed.) et al., Fractals in engineering. New trends in theory and applications. London: Springer (ISBN 1-84628-047-8/hbk; 1-84628-048-6/ebook). 199-217 (2005).
From the introduction: “The purpose of this paper is to give an overview of a class of convergence results for scaled random processes. The building blocks of the systems we study are real-valued standard random processes in continuous time, such as renewal, Lévy, or \(M/G/\infty \) models. Some higher-dimensional examples include the use of spatial Poisson point processes. The common feature is that all models involve heavy-tailed distributions, exhibit long-range dependence, and are naturally parameterized by the corresponding tail-index. To avoid an excessive number of similar models we focus on those that have stationary increments. We are interested in the stochastic fluctuations that build up when a large number of independent subsystems are super-positioned and simultaneously scaled in time (or, if applicable, in the space variable). In other words, our results concern convergence in distribution when performing double limits simultaneously. The proper scalings involve the tail-index and it turns out that there exist essentially three scaling regimes, each with a different asymptotic behavior. In Section 2 we begin with a brief presentation of six different models for which there are known results on the asymptotic behavior under double limit scaling. This includes super-positioned renewal counting processes, the sum of inverse Lévy subordinators, infinite source Poisson models, a self-similar rate model, a spatial interference model for wireless communications, and a spatial model for positively correlated mass configurations. In Section 3 we first discuss the three scaling regimes of fast, slow and intermediate growth, and give some heuristics for the corresponding limit processes. Then we state and compare the scaling limit results for the six different models. Section 4 contains some proofs that cannot be found elsewhere.”
For the entire collection see [Zbl 1083.00008].

60F05 Central limit and other weak theorems
60G51 Processes with independent increments; Lévy processes
60K05 Renewal theory
60K25 Queueing theory (aspects of probability theory)