A density property for stochastic processes

Abstract

Consider a class of probability distributions which is dense in the space of all probability distributions on $\mathbb{R}^{d}$ with respect to weak convergence, for every $d\in\mathbb{N}$. Then, we construct various explicit classes of continuous (c\'{a}dl\'{a}g) processes which are dense in the space of all continuous (c\'{a}dl\'{a}g) processes with respect to convergence in distribution. This is motivated by the recent result that quasi-infinitely divisible (QID) distributions are dense when $d=1$. If this result is extended to any $d\in\mathbb{N}$, then our result will imply that QID processes are dense in both spaces of continuous and c\'{a}dl\'{a}g processes.