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

A Logachov,A Yambartsev,A Mogulskii

doi:10.1051/ps/2022002

Abstract

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.

Highlights

We consider a class of chains r = ∈ AZ which is a special case of so-called chains with unbounded variable length memory or variable length Markov chain (VLMC)
These chains began to be actively studied after they were first introduced by Jorma Rissanen [22] as an economic and universal way of data compression
The short and simple introduction to these processes can be found, for example, in [14], see overview [7]. They are used for modeling data in computer science [22], in biology [1, 18], in neuro-biology [12, 13], in linguistics [15]

Summary

Introduction

We consider a class of chains r = (ri, i ∈ Z) ∈ AZ which is a special case of so-called chains with unbounded variable length memory or variable length Markov chain (VLMC). These chains began to be actively studied after they were first introduced by Jorma Rissanen [22] as an economic and universal way of data compression. Let us denote A−∞ the set of all such semi-infinite sequences. One call the process r(n) the VLMC if for any semi-infinite configuration α = {αi}−∞

LOGACHOV ET AL

Auxiliary results