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.
Keywords: Variable length memory chain, regeneration scheme, compound renewal process, local limit theorem, large deviation principle, moderate deviation principle, rate function, Cramer condition
@article{PS_2022__26_1_152_0,
author = {Logachov, A. and Mogulskii, A. and Yambartsev, A.},
title = {Limit theorems for chains with unbounded variable length memory which satisfy {Cramer} condition},
journal = {ESAIM: Probability and Statistics},
pages = {152--170},
year = {2022},
publisher = {EDP-Sciences},
volume = {26},
doi = {10.1051/ps/2022002},
mrnumber = {4382753},
zbl = {1485.60031},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ps/2022002/}
}
TY - JOUR AU - Logachov, A. AU - Mogulskii, A. AU - Yambartsev, A. TI - Limit theorems for chains with unbounded variable length memory which satisfy Cramer condition JO - ESAIM: Probability and Statistics PY - 2022 SP - 152 EP - 170 VL - 26 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/ps/2022002/ DO - 10.1051/ps/2022002 LA - en ID - PS_2022__26_1_152_0 ER -
%0 Journal Article %A Logachov, A. %A Mogulskii, A. %A Yambartsev, A. %T Limit theorems for chains with unbounded variable length memory which satisfy Cramer condition %J ESAIM: Probability and Statistics %D 2022 %P 152-170 %V 26 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/ps/2022002/ %R 10.1051/ps/2022002 %G en %F PS_2022__26_1_152_0
Logachov, A.; Mogulskii, A.; Yambartsev, A. Limit theorems for chains with unbounded variable length memory which satisfy Cramer condition. ESAIM: Probability and Statistics, Tome 26 (2022), pp. 152-170. doi: 10.1051/ps/2022002
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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.
