Some Open Probability Problems

This note presents four independent sets of open problems. The first set suggests an extension of the limit theory for positive recurrent renewal processes to the null 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.