Hit-and-run for numerical integration

We study the numerical computation of an expectation of a bounded function f with respect to a measure given by a non-normalized density on a convex body K ⊂ Rd . We assume that the density is log-concave, satisfies a variability condition and is not too narrow. In [19, 25, 26] it is required that K is the Euclidean unit ball. We consider general convex bodies or even the whole Rd and show that the integration problem satisfies a refined form of tractability. The main tools are the hit-and-run algorithm and an error bound of a multi run Markov chain Monte Carlo method.