Asymptotics of the Number of Different Words in a Markov Chain Driven Model

S. Fayzullaev, A. Kovalevskii

2025, v.31, Issue 3-4, 239-252

ABSTRACT

This paper investigates the asymptotic of the number of distinct words in a finite Markov chain driven model. We analyse the normalized and centered processes associated with the occurrence of distinct words in the model. Each state of the Markov chain is associated with its own unique infinite dictionary. At each state of the Markov chain, words are selected from the dictionary according to an infinite urn scheme. The probabilities in each infinite urn scheme satisfy the condition of regular variation. We use a combination of asymptotic techniques and results for Gaussian processes and derive the covariance structure of the limiting processes. The influence of stationary probabilities of the Markov chain on the normalization and scaling of these processes is explored in detail. Our findings provide new insights into the interaction between word frequencies and the stationary distribution in systems with pairwise disjoint
dictionaries. These results are applicable to a wide range of stochastic systems, offering a deeper understanding of their limiting behaviour.

doi:10.61102/1024-2953-mprf.2025.31.3-4.004

Keywords: Stationary distribution, Markov processes, Different words, Infinite urn scheme

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