A Bernstein-Type Inequality for Vector Functions on Finite Markov Chains

An analogue of the Bernstein inequality is derived for partial sums of a vector-valued function on a finite reversible Markov chain. The inequality gives an upper bound for the probability of a large deviation of the partial sum. The bound depends on the chain’s spectral gap, the dimension of the space where the function takes values, and the upper bound on the size and the variance of the function. 1 Result Let S be the state space of a finite Markov chain with transition matrix P and stationary distribution μ. Let the eigenvalues of P arranged in the declining order be λ1 = 1 > λ2 ≥ λ3 ≥ ... ≥ λ|S| > −1. Call the difference g = 1−λ2 the spectral gap of the chain. Recall that the chain is called reversible if μsPst = μtPts for any s and t from S. In this paper we discuss only reversible chains. Let f be a function on S that takes values in an m-dimensional real Hilbert space H. We use 〈, 〉 and |·| to denote the scalar product and norm in this Hilbert space. We study the behavior of the norm of partial sums SN = ∑N t=1 f(st). [email protected] 1 The behavior of SN depends on the initial distribution μ (0) and properties of the function f . Define the following distance: