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We present a categorical viewpoint of probability measures by showing that a probability measure can be viewed as a weakly averaging affine measurable functional taking values in the unit interval which preserves limits. The probability…
Based on the probability distribution observed in complex systems and an assumption that the probability distributions of complex systems satisfy a new generalized multiplication, it is proved that the statistical theory of complex systems…
Quantum systems are the ultimate touchstone for the production of random sequences of numbers. Spatially spread entangled systems allow the generation of identical random sequences in remote locations. The impossibility of observing a…
This paper constructs a solvability theory for a system of stochastic partial differential equations. On account of the Kolmogorov continuity theorem, solutions are looked for in certain H\"older-type classes in which a random field is…
Mutual information I in infinite sequences (and in their finite prefixes) is essential in theoretical analysis of many situations. Yet its right definition has been elusive for a long time. I address it by generalizing Kolmogorov Complexity…
Let E be a locally compact second countable Hausdorff space and F the pertaining family of all closed sets. We endow F respectively with the Fell-topology, the upper Fell topology or the upper Vietoris-topology and investigate weak…
We investigate the power of weak measurements in the framework of quantum state discrimination. First, we define and analyze the notion of weak consecutive measurements. Our main result is a convergence theorem whereby we demonstrate when…
In this paper, we characterize compatibility of distributions and probability measures on a measurable space. For a set of indices $\mathcal J$, we say that the tuples of probability measures $(Q_i)_{i\in \mathcal J} $ and distributions…
We propose a finite volume stochastic collocation method for the random Euler system. We rigorously prove the convergence of random finite volume solutions under the assumption that the discrete differential quotients remain bounded in…
We define the empiric stochastic stability of an invariant measure in the finite-time scenario, the classical definition of stochastic stability. We prove that an invariant measure of a continuous system is empirically stochastically stable…
The paper is devoted to studying the image of probability measures on a Hilbert space under finite-dimensional analytic maps. We establish sufficient conditions under which the image of a measure has a density with respect to the Lebesgue…
We use the fact that some linear Hamiltonian systems can be considered as ``finite level'' quantum systems, and the description of quantum mechanics in terms of probabilities, to associate probability distributions with this particular…
We establish various forms of the following certainty principle: a set $S \subset \mathbb{R}^{n}$ contains a given finite linear pattern, provided that $S$ is a support of the Fourier transform of a sufficiently singular probability measure…
We first introduce the concept of weak random periodic solutions of random dynamical systems. Then, we discuss the existence of such periodic solutions. Further, we introduce the definition of weak random periodic measures and study their…
We provide a characterization in terms of Fatou closedness for weakly closed monotone convex sets in the space of $\mathcal{P}$-quasisure bounded random variables, where $\mathcal{P}$ is a (possibly non-dominated) class of probability…
We describe a mathematical language for determining all possible patterns of contextuality in the dependence of stochastic outputs of a system on its deterministic inputs. The central notion is that of all possible couplings for…
We show that for a convex solid set of positive random variables to be tight, or equivalently bounded in probability, it is necessary and sufficient that it is radially bounded, i.e. that every ray passing through one of its elements…
The set of subsystems of a finite quantum system (with variables in Z(n)) together with logical connectives, is a distributive lattice. With regard to this lattice, the (where P(m) is the projector to) obeys a supermodularity inequality,…
We present an algorithm that takes a discrete random variable $X$ and a number $m$ and computes a random variable whose support (set of possible outcomes) is of size at most $m$ and whose Kolmogorov distance from $X$ is minimal. In addition…
Weak convergence of probability measures is one of the most important topics in the field probability and statistics. In this survey paper, we look at weak convergence of probability measures from the topological vector space point of view.…