Open problems in renewal, coupling and Palm theory

Abstract

This note presents four sets of problems. The first suggests the possibility of a limit theory for null-recurrent renewal processes similar to the theory in the positive recurrent case. The second concerns exact coupling of random walks on the line with step-lengths that are neither discrete nor spread out. The third concerns the coupling characterization of setwise convergence of distributions of stochastic processes to a stationary limit. The fourth concerns characterizations of mass-stationarity, a concept formalizing the intuitive idea that the origin is a typical location in the mass of a random measure. Mass-stationarity is an intrinsic characterization of Palm versions with respect to stationary random measures.