# Hoeffding's lemma for Markov Chains and its applications to statistical learning

@article{Fan2018HoeffdingsLF, title={Hoeffding's lemma for Markov Chains and its applications to statistical learning}, author={Jianqing Fan and Bai Jiang and Qiang Sun}, journal={arXiv: Statistics Theory}, year={2018} }

We establish the counterpart of Hoeffding's lemma for Markov dependent random variables. Specifically, if a stationary Markov chain $\{X_i\}_{i \ge 1}$ with invariant measure $\pi$ admits an $\mathcal{L}_2(\pi)$-spectral gap $1-\lambda$, then for any bounded functions $f_i: x \mapsto [a_i,b_i]$, the sum of $f_i(X_i)$ is sub-Gaussian with variance proxy $\frac{1+\lambda}{1-\lambda} \cdot \sum_i \frac{(b_i-a_i)^2}{4}$. The counterpart of Hoeffding's inequality immediately follows. Our results…

## 27 Citations

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