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Register a new userDenseness of adapted processes among causal couplings
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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.
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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. 4.1]{lindner2018} it was shown that the class of quasi-infinitely divisible distributions on $mathbb{R}$ is dense in the class of distributions on $mathbb{R}$ with respect to weak convergence. In this paper, we show that the class of quasi-infinitely divisible distributions on $mathbb{R}^d$ is not dense in the class of distributions on $mathbb{R}^d$ with respect to weak convergence if $d geq 2$.
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 to obtain uniform error bounds for generalized gamma approximations using Steins method. Here we show how monotone couplings arising with these fixed point equations can be used to obtain sharper tail bounds that, in many cases, outperform competing moment-based bounds and the uniform bounds obtainable with Steins method. Applications are given to concentration inequalities for preferential attachment random graphs, branching processes, random walk local time statistics and the size of random subtrees of uniformly random binary rooted plane trees.
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 variable selection approach is specially designed for causal inference, it only requires the propensity scores to be $sqrt{n}$-consistently estimated through a parametric model and need not correct specification of potential outcome models. By using estimated propensity scores as inverse probability treatment weights in performing an adaptive lasso on the outcome, it successfully excludes instrumental variables, and includes confounders and outcome predictors. We show its oracle properties under the linear association conditions. We also perform some numerical simulations to illustrate our propensity score adapted covariate selection procedure and evaluate its performance under model misspecification. Comparison to other covariate selection methods is made using artificial data as well, through which we find that it is more powerful in excluding instrumental variables and spurious covariates.
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 physical contact with another particle for a finite period of time. We refer to this process as a reaction process and to the different internal states as particle species making an analogy to chemical reactions. The studied system may fall into an absorbing phase, where due to the disappearance of one of the particle species no further reaction can occur or remain in an active phase where particles constantly react. Combining individual-based simulations and mean-field arguments, we study the dependence of the equilibrium densities of particle species with motility parameters, specifically the active speed $v_0$ and tumbling frequency $lambda$. We find that the equilibrium densities of particle species exhibit two very distinct, non-trivial scaling regimes with $v_0$ and $lambda$ depending on whether the system is in the so-called ballistic or diffusive regime. Our mean-field estimates lead to an effective renormalization of reaction rates that allow building the phase-diagram $v_0$--$lambda$ that separates the absorbing and active phase. We find an excellent agreement between numerical simulations and estimates. This study is a necessary step to an understanding of phase transitions into an absorbing state in active systems and sheds light on the spreading of information/signaling among moving elements.
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 assumption of causal order between them. We consider the general multipartite case and take into account the possibility for dynamical causal order, where the order of a set of events can depend on other events in the past. Starting from a general definition of causality, we derive an iteratively formulated canonical decomposition of multipartite causal processes, and show that for a fixed number of settings and outcomes for each party, the respective correlations form a polytope whose facets define causal inequalities. In the case of quantum processes, we investigate the link between causality and the theory-dependent notion of causal separability, which we here extend to the multipartite case based on concrete principles. We show that causality and causal separability are not equivalent in general by giving an example of a physically admissible tripartite quantum process that is causal but not causally separable. We also show that there exist causally separable (and hence causal) quantum processes that become non-causal if extended by supplying the parties with entangled ancillas. This example of activation of non-causality motivates the concepts of extensibly causal and extensibly causally separable (ECS) processes, for which the respective property remains invariant under extension with arbitrary ancillas. We characterize the class of tripartite ECS processes in terms of simple conditions on the form of the process matrix, which generalize the form of bipartite causally separable process matrices. We show that the processes realizable by classically controlled quantum circuits are ECS and conjecture that the reverse also holds.
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