Stochastic chains with unbounded memory are a natural generalization of Markov chains, in the sense that the transition probabilities may depend on the whole past. These process, introduced by Onicescu and Mihoc (1935) and Doeblin and Fortet (1937), have been receiving increasing attention in the probabilistic literature, because they form a class richer than the Markov chains and have practical capabilities modelling of scientific data in several areas, from biology to linguistics. In this work, we intend to use them to model properties of brain activity such as spike trains. The main objective in this project is to develop new mathematical results about stochastic chains with unbounded memory. First, we intend to study the sufficient conditions that guarantee the existence and uniqueness of invariant measure for such chains and on this occasion, addressing issues such as law of large numbers (ergodicity of chain), central limit theorem, large deviations and concentration inequality. Then, we will address the understanding of the phenomenology of spike trains and their modelling via stochastic chains with unbounded memory.
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