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

2024-10-07

arXiv.org

AI Summary

This paper addresses the weak statistical foundations of nonlinear time series models in econometrics, queueing theory, and machine learning, specifically focusing on (i) how α-mixing of exogenous regressors propagates to the response variable, and (ii) mixing preservation and limiting behavior of nonstationary Markov chains under stochastic nonstationarity. We develop a rigorous propagation mechanism based on first-passage coupling and establish an analytical framework for nonstationary Markov chains incorporating drift conditions and small-set structures. Departing from conventional stationarity assumptions, we derive, for the first time in nonstationary stochastic environments, a strong law of large numbers (LLN) and a functional central limit theorem (FCLT) for weakly dependent nonlinear sequences. Our results yield novel theoretical foundations for asymptotic normality and provide a stability criterion with guaranteed mixing properties for single-server queueing systems.

Technology Category

Machine Learning: Other Foundations of Machine LearningReasoning under Uncertainty: Stochastic OptimizationSearch and Optimization: Non-convex Optimization

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSecurity and Privacy: Large-scale security measurementsWeb Mining and Content Analysis: Models for Web evolution

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.

Problem

• Analyzing mixing properties in nonlinear time series with exogenous regressors
• Extending limit theorems to non-stationary random environment Markov chains
• Applying mixing transitions to queuing theory for single-server models

Innovation

• Transfer mixing properties via coupling arguments
• Study Markov chains in random environments
• Apply framework to single-server queuing models