A large deviation inequality for vector functions on finite reversible Markov Chains

Let SN be the sum of vector-valued functions defined on a finite Markov chain. An analogue of the Bernstein.Hoeffding inequality is derived for the probability of large deviations of SN and relates the probability to the spectral gap of the Markov chain. Examples suggest that this inequality is better than alternative inequalities if the chain has a sufficiently large spectral gap and the function is high-dimensional.