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Convergence of Markov Chains to self-similar Processes

Self-similar processes are stochastic processes that are invariant in distribution under suitable scaling of time and space. These processes can be used to model many space-time scaling random phenomena that can be observed in physics, biology and other fields. One could mention stellar fragments, growth and genealogy of populations, option pricing in finance, various areas of image processing, climatology, environmental science, . . . Self-similar processes appear in various parts of probability theory, such as in Lévy processes, branching processes, statistical physics, fragmentation theory, random fields, … Some well known examples are: stable Lévy process, fractional Brownian motion,. . . but it has also been shown that relatively simple Markov models can produce self-similarity. Even though the cardinality of the state space increases to infinity, it has also been shown that its rate is quite low. The aim of the study is to prove how theses Markov models converge to self-similar limit processes, under which conditions.