Expectations for nonreversible Markov chains (Q1270857)

The paper is concerned with a Markov chain \(X=(X_n, n\geq 0)\), taking values in a compact separable metric space \(S\) and let \(P(x,A)\) be its transitional probability which is supposed to be irreducible, i.e., for each upon set \(U\subset S\) and each \(x\in S\), there exists \(k\geq 1\) such that \(P^k(x, U)>0\). Let \(\mu\) be the invariant measure for \(P(\cdot,\cdot)\) and \(f\) be the continuous mapping of \(S\) into \([0,1]\). Denote \(\mu_f= \int_Sf(x) \mu(dx)\) and let \({\mathbf P}_x\) signify the conditional probability provided \(X_0=x\). Let the operator \(P\) be generated by the kernel \(P(\cdot,\cdot)\) by the usual way: \(Pf (x) =\int_Sf(y) P(x,dy)\). The author is interested in obtaining bounds uniform in the staring point \(x\) for \({\mathbf P}_x\{n^{-1} \sum^n_{k=1} f(X_k)-\mu_f\geq \varepsilon\}\). The main result is as follows: \[ \inf_x{\mathbf P}_x \left\{n^{-1} \sum^n_{k=1} f(X_k)-\mu_f \geq\varepsilon \right\}\leq \exp\bigl\{-n \beta \varepsilon^2(1- \varepsilon)2^8 \bigr\}, \] where \(\beta\) is, roughly speaking, the square root of the distance between zero and the remainder part of the spectrum of the operator \(P-I\).