TY - JOUR

T1 - Asymptotics of the Number of Different Words in a Markov Chain Driven Model

AU - Fayzullaev, Shahzod

AU - Kovalevskii, Artyom

N1 - FWNF-2026-0030

PY - 2025/4/25

Y1 - 2025/4/25

N2 - 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 dictio-nary. 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 struc-ture 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.

AB - 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 dictio-nary. 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 struc-ture 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.

KW - Different words

KW - Infinite urn scheme

KW - Markov processes

KW - Stationary distribution

UR - https://www.scopus.com/pages/publications/105037409268

UR - https://www.mendeley.com/catalogue/f3bb945d-5920-3ecc-ba50-97a30b8234fa/

U2 - 10.61102/1024-2953-mprf.2025.31.3-4.004

DO - 10.61102/1024-2953-mprf.2025.31.3-4.004

M3 - Article

VL - 31

SP - 239

EP - 252

JO - Markov Processes And Related Fields

JF - Markov Processes And Related Fields

SN - 1024-2953

IS - 3-4

ER -