Markov Sets.

Regenerative embedding of Markov sets

Given a closed Markov (i.e. regenerative) set in ‰0;1†, we characterize the laws of the Markov sets which are regeneratively embedded into the latter. Typically, let U and U be two Laplace exponents corresponding to two regenerative laws, and M …2† a Markov set with exponent U . There exists a Markov set M …1† with exponent U which is regeneratively embedded into M …2† if and only if U=U is a c...

Small and Pseudo - Small Sets for Markov

Small sets and Markov transition densities

The theory of general state-space Markov chains can be related strongly to the case of discrete state-space by use of Doeblin’s notion of small sets and associated minorization conditions. The general theory shows that small sets exist for all Markov chains on state-spaces with countably generated -algebras, though the minorization provided by the theory concerns n-step transition kernels for s...

On Markov Chains for Independent Sets

Random independent sets in graphs arise, for example, in statistical physics, in the hard-core model of a gas. In 1997, Luby and Vigoda described a rapidly mixing Markov chain for independent sets, which we refer to as the Luby–Vigoda chain. A new rapidly mixing Markov chain for independent sets is defined in this paper. Using path coupling, we obtain a polynomial upper bound for the mixing tim...

Turán type reverse Markov inequalities for compact convex sets

For a compact convex set K ⊂ C the well-known general Markov inequality holds asserting that a polynomial p of degree n must have ∥∥p′∥∥ c(K)n2 ‖p‖. On the other hand for polynomials in general, ∥∥p′∥∥ can be arbitrarily small as compared to ‖p‖. The situation changes when we assume that the polynomials in question have all their zeroes in the convex set K. This was first investigated by Turán,...