A Denseness Property of Stochastic Processes

Abstract In this work, we show the following general property of stochastic processes: Denseness of probability distributions extends to denseness of stochastic processes. In particular, for every $$d\in \mathbb {N}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> , we consider any class of probability distributions which is dense in the space of probability distributions on $$\mathbb {R}^{d}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> with respect to convergence in distribution. Using these classes, we construct various explicit families of times series, continuous processes and càdlàg processes that are dense in their respective spaces with respect to convergence in distribution. This is particularly interesting because, for processes, finite-dimensional distributions convergence does not automatically imply convergence in distribution.