Periodically homogeneous Markov chains: The discrete state space case

Aknouche, Abdelhakim (2024): Periodically homogeneous Markov chains: The discrete state space case.

Abstract

A unified theory of periodically homogeneous Markov chains on countable state spaces with periodically time-varying transition probabilities is introduced. The finite-dimensional probability distributions of these time-periodic chains are first studied and their correspondence with the marginal distributions and transition probabilities is shown. Then, the concepts of periodic stability/regularity and limiting behaviors are proposed. The communicability and classification of states necessary for establishing periodic stability are then examined. In particular, periodic irreducibility and the main solidarity/class properties are presented, namely periodic recurrence, periodic positive recurrence, periodic transience, and periodic aperiodicity. Furthermore, sufficient conditions for periodic stochastic stability of time-periodic Markov chains are derived. Finally, various applications to some operations research models and time series analysis are considered. In particular, periodic Markov decision processes, periodic integer-valued time series models, and periodic Markov-switching time series models are examined.

• Item Type: MPRA Paper
• Language: English
• Keywords: Time-periodic Markov chains, Harris periodic ergodicity, periodic irreducibility, periodic recurrence, periodic stability, periodic invariant distributions, periodic integer-valued time series models, Markov-switching periodic models, periodic Markov decision process.
• Subjects: C - Mathematical and Quantitative Methods > C0 - General > C01 - Econometrics; C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods; C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables > C30 - General

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