A Denseness Property of Stochastic Processes

Riccardo Passeggeri ()
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Riccardo Passeggeri: Imperial College London

Journal of Theoretical Probability, 2025, vol. 38, issue 1, 1-18

Abstract: 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 $$d\in \mathbb {N}$$ d ∈ N , we consider any class of probability distributions which is dense in the space of probability distributions on $$\mathbb {R}^{d}$$ R d 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.

Keywords: Time series; Stochastic processes; Convergence in distribution; Weak convergence; Denseness; Approximating class; 60B10; 60G07; 60G05; 62H05 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10959-024-01372-y

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