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Transition of $α$-mixing in Random Iterations with Applications in Queuing Theory

Attila Lovas

TL;DR

This work develops a coupling-based approach to transfer $\alpha$-mixing from exogenous regressors to nonlinear time-series recursions, enabling LLN and CLT results for functionals even in nonstationary environments. It introduces a Markov chain in random environment (MCRE) framework with drift and minorization, proving LLN and functional CLTs under favorable environment mixing, and establishing forward coupling to a stationary surrogate with explicit TV-rate bounds. The results extend to queuing models, where waiting times follow the Lindley recursion and can be analyzed as MCREs with dependent inputs, yielding quantitative convergence rates and functional limit theorems under various mixing assumptions. Collectively, the paper provides a rigorous probabilistic foundation for statistical inference in nonlinear autoregressive models with exogenous covariates and in queuing systems with complex dependency structures, including nonstationary data streams.

Abstract

Nonlinear time series models with exogenous regressors are essential in econometrics, queuing theory, and machine learning, though their statistical analysis remains incomplete. Key results, such as the law of large numbers and the functional central limit theorem, are known for weakly dependent variables. We demonstrate the transfer of mixing properties from the exogenous regressor to the response via coupling arguments. Additionally, we study Markov chains in random environments with drift and minorization conditions, even under non-stationary environments with favorable mixing properties, and apply this framework to single-server queuing models.

Transition of $α$-mixing in Random Iterations with Applications in Queuing Theory

TL;DR

This work develops a coupling-based approach to transfer -mixing from exogenous regressors to nonlinear time-series recursions, enabling LLN and CLT results for functionals even in nonstationary environments. It introduces a Markov chain in random environment (MCRE) framework with drift and minorization, proving LLN and functional CLTs under favorable environment mixing, and establishing forward coupling to a stationary surrogate with explicit TV-rate bounds. The results extend to queuing models, where waiting times follow the Lindley recursion and can be analyzed as MCREs with dependent inputs, yielding quantitative convergence rates and functional limit theorems under various mixing assumptions. Collectively, the paper provides a rigorous probabilistic foundation for statistical inference in nonlinear autoregressive models with exogenous covariates and in queuing systems with complex dependency structures, including nonstationary data streams.

Abstract

Nonlinear time series models with exogenous regressors are essential in econometrics, queuing theory, and machine learning, though their statistical analysis remains incomplete. Key results, such as the law of large numbers and the functional central limit theorem, are known for weakly dependent variables. We demonstrate the transfer of mixing properties from the exogenous regressor to the response via coupling arguments. Additionally, we study Markov chains in random environments with drift and minorization conditions, even under non-stationary environments with favorable mixing properties, and apply this framework to single-server queuing models.
Paper Structure (6 sections, 27 theorems, 205 equations, 1 figure)

This paper contains 6 sections, 27 theorems, 205 equations, 1 figure.

Table of Contents

  1. Transition of mixing properties
  2. Markov chains in random environments
  3. Single server queuing systems
  4. Brief survey of key results on $\alpha$-mixing sequences
  5. Counterexample to long-term contractivity condition
  6. Coupling condition for MCREs

Key Result

Lemma 1.2

Assume that $X_0$ is a random initial state independent of $\sigma (Y_n,\varepsilon_{n+1}\mid n\in\mathbb{N})$, moreover the iteration eq:iter satisfies the coupling condition with $x_0\in\mathcal{X}$. Then for $0\le m<n$, we have where $b(n) = \sup_{j\in\mathbb{N}}\mathbb{P} \left(Z_{j,j+n}^{X_j}\ne Z_{j,j+n}^{x_0}\right).$

Figures (1)

  • Figure 1: Schematic overview of the single-server queuing system under investigation: $Z_n$ is the time between the arrivals of the $n$-th and $(n+1)$-th customers, $S_n$ is the time to serve the $n$-th customer, and $W_n$ is their waiting time before service.

Theorems & Definitions (64)

  • Definition 1.1
  • Lemma 1.2
  • proof
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • proof
  • Definition 2.1
  • Lemma 2.3
  • proof
  • ...and 54 more