We extend Hoeffdings lemma to general-state-space and not necessarily reversible Markov chains. Let ${X_i}_{i ge 1}$ be a stationary Markov chain with invariant measure $pi$ and absolute spectral gap $1-lambda$, where $lambda$ is defined as the operator norm of the transition kernel acting on mean zero and square-integrable functions with respect to $pi$. 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 $frac{1+lambda}{1-lambda} cdot sum_i frac{(b_i-a_i)^2}{4}$. This result differs from the classical Hoeffdings lemma by a multiplicative coefficient of $(1+lambda)/(1-lambda)$, and simplifies to the latter when $lambda = 0$. The counterpart of Hoeffdings inequality for Markov chains immediately follows. Our results assume none of countable state space, reversibility and time-homogeneity of Markov chains and cover time-dependent functions with various ranges. We illustrate the utility of these results by applying them to six problems in statistics and machine learning.
تحميل البحث