A Reciprocity Theorem for Ergodic Actions

A Ratio Ergodic Theorem for Multiparameter Non-singular Actions

We prove a ratio ergodic theorem for non-singular free Z and R actions, along balls in an arbitrary norm. Using a Chacon-Ornstein type lemma the proof is reduced to a statement about the amount of mass of a probability measure that can concentrate on (thickened) boundaries of balls in R. The proof relies on geometric properties of norms, including the Besicovitch covering lemma and the fact tha...

On the ratio ergodic theorem for group actions

We study the ratio ergodic theorem (RET) of Hopf for group actions. Under a certain technical condition, if a sequence of sets {Fn} in a group satisfy the RET, then there is a finite set E such that {EFn} satisfies the Besicovitch covering property. Consequently for the abelian group G = ⊕n=1Z there is no sequence Fn ⊆ G along which the RET holds, and in many finitely generated groups, includin...

A Marginal Ergodic Theorem

Michael Lavine Duke University, Durham, NC, USA. Summary. In recent years there have been several papers giving examples of Markov Chain Monte Carlo (MCMC) algorithms whose invariant measures are improper (have infinite mass) and which therefore are not positive recurrent, yet which have subchains from which valid inference can be derived. These are nonergodic (not having a limiting distributio...

A Mean Ergodic Theorem For Asymptotically Quasi-Nonexpansive Affine Mappings in Banach Spaces Satisfying Opial's Condition

A Combinatorial Reciprocity Theorem for Hyperplane Arrangements

Given a nonnegative integer m and a finite collection A of linear forms on Qd, the arrangement of affine hyperplanes in Qd defined by the equations α(x) = k for α ∈ A and integers k ∈ [−m,m] is denoted by Am. It is proved that the coefficients of the characteristic polynomial of Am are quasi-polynomials inm and that they satisfy a simple combinatorial reciprocity law.