Abstract: This paper investigates the asymptotic of the number of distinct words in a nite 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 innite dictionary. At each state of the Markov chain, words are selected from the dictionary according to an innite urn scheme. The probabilities in each innite 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 inuence of stationary probabilities of the Markov chain on the normalization and scaling of these processes is explored in detail. Our ndings 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, o ering a deeper understanding of their limiting behaviour.

Cite: Fayzullaev S. , Kovalevskii A.
Asymptotics of the Number of Different Words in a Markov Chain Driven Model
Markov Processes and Related Fields. 2026. V.31. N3-4 (2025). P.239-252. DOI: 10.61102/1024-2953-mprf.2025.31.3-4.004 Scopus OpenAlex

Dates:

Submitted:	Oct 25, 2025
Accepted:	Mar 11, 2026
Published online:	Apr 25, 2026

Identifiers:

≡ Scopus:	2-s2.0-105037409268
≡ OpenAlex:	W7155395355

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