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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 $\\frac{1+\\lambda}{1-\\lambda} \\cdot \\sum_i \\frac{(b_i-a_i)^2}{4}$. 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