Volume 26, 2022
|Page(s)||152 - 170|
|Published online||21 February 2022|
Limit theorems for chains with unbounded variable length memory which satisfy Cramer condition*
Laboratory of Probability Theory and Mathematical Statistics, Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Koptuga, 4,
630090, Russian Fedration.
2 Department of High Mathematics, Siberian State University of Geosystems and Technologies, Plahotnogo str. 10, Novosibirsk 630108, Russian Fedration.
3 Department of Statistics, Institute of Mathematics and Statistics (IME-USP), University of São Paulo, Rua do Matão 1010, CEP 05508–090, São Paulo SP, Brazil.
** Corresponding author: email@example.com
Accepted: 5 February 2022
We consider a class of variable length Markov chains with a binary alphabet in which context tree is defined by adding finite trees with uniformly bounded height to the vertices of an infinite comb tree. Such type of Markov chain models the spike neuron patterns and also extends the class of persistent random walks. The main interest is the limiting properties of the empirical distribution of symbols from the alphabet. We obtain the strong law of large numbers, central limit theorem, and exact asymptotics for large and moderate deviations. The presence of an intrinsic renewal structure is the subject of discussion in the literature. Proofs are based on the construction of a renewals of the chain and the applying corresponding properties of the compound (or generalized) renewal processes.
Mathematics Subject Classification: 60F10 / 60K05 / 60K30
Key words: Variable length memory chain / regeneration scheme / compound renewal process / local limit theorem / large deviation principle / moderate deviation principle / rate function / Cramer condition
This work was supported by the Basic Research Program of the Siberian Branch of the Russian Academy of Sciences, project No. FWNF-2022-0010. It is part of USP project Mathematics, computation, language and the brain, FAPESP project Research, Innovation and Dissemination Center for Neuromathematics grant 2013/07699-0. AL and AY thank FAPESP grant 2017/20482-0, AY also thanks CNPq and FAPESP grants 301050/2016-3 and 2017/10555-0, respecively.
© The authors. Published by EDP Sciences, SMAI 2022
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