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2019 On the existence of continuous processes with given one-dimensional distributions
Luca Pratelli, Pietro Rigo
Electron. Commun. Probab. 24(none): 1-9 (2019). DOI: 10.1214/19-ECP255


Let $\mathcal{P} $ be the collection of Borel probability measures on $\mathbb{R} $, equipped with the weak topology, and let $\mu :[0,1]\rightarrow \mathcal{P} $ be a continuous map. Say that $\mu $ is presentable if $X_{t}\sim \mu _{t}$, $t\in [0,1]$, for some real process $X$ with continuous paths. It may be that $\mu $ fails to be presentable. Hence, firstly, conditions for presentability are given. For instance, $\mu $ is presentable if $\mu _{t}$ is supported by an interval (possibly, by a singleton) for all but countably many $t$. Secondly, assuming $\mu $ presentable, we investigate whether the quantile process $Q$ induced by $\mu $ has continuous paths. The latter is defined, on the probability space $((0,1),\mathcal{B} (0,1),\mbox{Lebesgue measure} )$, by \[ Q_{t}(\alpha )=\inf \, \bigl \{x\in \mathbb{R} :\mu _{t}(-\infty ,x]\ge \alpha \bigl \} \quad \quad \mbox{for all } t\in [0,1]\mbox{ and } \alpha \in (0,1). \] A few open problems are discussed as well.


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Luca Pratelli. Pietro Rigo. "On the existence of continuous processes with given one-dimensional distributions." Electron. Commun. Probab. 24 1 - 9, 2019.


Received: 12 June 2019; Accepted: 2 July 2019; Published: 2019
First available in Project Euclid: 28 August 2019

zbMATH: 07107310
MathSciNet: MR4003120
Digital Object Identifier: 10.1214/19-ECP255

Primary: 60A05, 60B10, 60G05, 60G17


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