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Limit theorems for a higher order time dependent Markov chain model

Kokoszka Piotr, Kutta Tim, Singh Deepak, Wang Haonan

Probability and Mathematical Statistics

2023

Vol. 43, Fasc. 1

121--139

The paper establishes a strong law of large numbers and a central limit theorem for a sequence of dependent Bernoulli random variables modeled as a higher order Markov chain. The model under consideration is motivated by problems in quality control where acceptability of an item depends on the past k acceptability scores. Moreover, the model introduces dependence that may evolve over time and thus advances the theory for models with time invariant dependence. We establish explicit assumptions that incorporate this dynamic dependence and show how it enters into the limits describing long-term behavior of the system.