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It is well known that any pair of random variables $(X,Y)$ with values in Polish spaces, provided that $Y$ is nonatomic, can be approximated in joint law by random variables of the form $(X,Y)$ where $X$ is $Y$-measurable and $X stackrel{d}{=} X$. This article surveys and extends some recent dynamic analogues of this result. For example, if $X$ and $Y$ are stochastic processes in discrete or continuous time, then, under a nonatomic assumption as well as a necessary and sufficient causality (or compatibility) condition, one can approximate $(X,Y)$ in law in path space by processes of the form $(X,Y)$, where $X$ is adapted to the filtration generated by $Y$. In addition, in finite discrete time, we can take $X$ to have the same law as $X$. A similar approximation is valid for randomized stopping times, without the first marginal fixed. Natural applications include relaxations of (mean field) stochastic control and causal optimal transport problems as well as new characterizations of the immersion property for progressively enlarged filtrations.
A probability distribution $mu$ on $mathbb{R}^d$ is quasi-infinitely divisible if its characteristic function has the representation $widehat{mu} = widehat{mu_1}/widehat{mu_2}$ with infinitely divisible distributions $mu_1$ and $mu_2$. In cite[Thm. 4
Generalized gamma distributions arise as limits in many settings involving random graphs, walks, trees, and branching processes. Pekoz, Rollin, and Ross (2016, arXiv:1309.4183 [math.PR]) exploited characterizing distributional fixed point equations t
In this paper, we propose a propensity score adapted variable selection procedure to select covariates for inclusion in propensity score models, in order to eliminate confounding bias and improve statistical efficiency in observational studies. Our v
We study a system of self-propelled disks that perform run-and-tumble motion, where particles can adopt more than one internal state. One of those internal states can be transmitted to another particle if the particle carrying this state maintains ph
We develop rigorous notions of causality and causal separability in the process framework introduced in [Oreshkov, Costa, Brukner, Nat. Commun. 3, 1092 (2012)], which describes correlations between separate local experiments without a prior assumptio