Limit theorems for chains with unbounded variable length memory which satisfy Cramer condition

被引:0
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作者
Logachov, A. [1 ,2 ]
Mogulskii, A. [1 ]
Yambartsev, A. [3 ]
机构
[1] Laboratory of Probability Theory and Mathematical Statistics, Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Koptuga, 4, Novosibirsk,630090, Russia
[2] Department of High Mathematics, Siberian State University of Geosystems and Technologies, Plahotnogo str. 10, Novosibirsk,630108, Russia
[3] Department of Statistics, Institute of Mathematics and Statistics (IME-USP), University of São Paulo, Rua do Matão 1010, SP, São Paulo,CEP 05508-090, Brazil
基金
巴西圣保罗研究基金会;
关键词
Compound renewal process - Condition - Crame condition - Large deviation principle - Local limit theorem - Memory chain - Moderate deviation principle - Rate functions - Regeneration scheme - Renewal process - Variable length - Variable length memory chain;
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摘要
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. © The authors. Published by EDP Sciences SMAI 2022.
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页码:152 / 170
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