Hoeffding's inequality for general Markov chains with its applications to statistical learning.

Area of Science:

• Probability Theory
• Statistical Inference
• Markov Chain Analysis

Background:

• Hoeffding's inequality is fundamental for independent random variables.
• Generalizing these bounds to dependent data, like Markov chains, is crucial for complex systems.
• Existing inequalities often require strong assumptions like reversibility or specific state spaces.

Purpose of the Study:

• To establish Hoeffding's lemma and inequality for general-state-space, non-reversible Markov chains.
• To characterize the sharpness of these inequalities by comparing variance proxies.
• To demonstrate the necessity of function boundedness for these generalized results.

Main Methods:

• Development of Hoeffding-type bounds tailored for Markov chains.
• Analysis of variance proxies to establish the optimality of the derived inequalities.
• Theoretical proofs demonstrating the necessity of function boundedness.

Main Results:

• Hoeffding's lemma and inequality are established for a broad class of Markov chains.
• The sharpness of these bounds is quantified through the ratio of variance proxies.
• Boundedness of functions is proven to be a necessary condition for the general validity of these inequalities.

Conclusions:

• The established inequalities provide powerful tools for non-asymptotic analysis in various fields.
• The results offer new theoretical underpinnings for MCMC estimation, respondent-driven sampling, and time series analysis.
• Applications extend to econometrics and machine learning, including Markovian reward multi-armed bandit problems.