Hoeffding's inequality for Markov processes via solution of Poisson's equation
The authors investigate Hoeffding's inequality for both discrete-time Markov chains and continuous-time Markov processes on a general state space. In the words of the authors, ``Our results relax the usual aperiodicity restriction in the literature, and the explicit upper bounds in the inequalities are obtained via the solution of Poisson's equation. The results are further illustrated with applications to queueing theory and reflective diffusion processes.'' The classical Hoeffding's inequality gives an exponentially decaying upper bound to the sample mean of a sequence of independent and identically distributed and bounded random variables. Now, for any bounded real-valued function \(g\), and \(X_n\) a discrete time positive recurrent Markov chain with invariant distribution \(\pi\), one defines \[ S_n(g)=\sum_{i=0}^{n-1}g(X_i)\,, \] and the Hoeffding's inequality in this case gives an upper bound, that decays exponentially in \(n\), to the probability that \(\frac{1}{n}S_n(g)-\pi(g)\ge \varepsilon\). Such an inequality has a lot of applications in probability and statistics. The authors modify the arguments in [\textit{P. W. Glynn} and \textit{D. Ormoneit}, Stat. Probab. Lett. 56, No. 2, 143--146 (2002; Zbl 0999.60019)] to derive a Hoeffding's inequality for discrete time Markov chains by bounding the solution of Poisson's equation through the drift condition and ergodicity coefficient. For continuous time Markov processes, they extend the arguments of \textit{M. C. H. Choi} and \textit{E. Li} [Stat. Probab. Lett. 150, 23--28 (2019; Zbl 1448.60047)] to derive a Hoeffding's inequality by bounding the solution of Poisson's equation in terms of the hitting times and the drift condition.
scientific article; zbMATH DE number 7379090
scientific article