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For the set I (X) of probability measures on a compact Hausdorff space X, we propose a new way to introduce the topology by using the open subsets of the space X. Then, among other things, we give a new proof that for a compact Hausdorff space X, the space I (X) is also a compact Hausdorff space. For a Tychonoff space X, we consider the topological space I τ (X) of τ -smooth idempotent probability measures on X and show that the space I τ (X) is Čech-complete if and only if the given space X is Čech-complete. [ABSTRACT FROM AUTHOR]

In this paper, we show that a sufficient and necessary condition for that the Hoeffding concentration inequality of Banach space-valued (B -valued, for simplicity) random variables holds is that the Banach space in question admits the Rademacher type p for some 1 ≤ p ≤ 2 ; which is equivalent to that the Bernstein concentration inequality holds for such B -valued random variables. This is done by applying a refined McDiarmid inequality (stated and proved in this paper) via the entropy method. This result is also applied to the density property of induced subgraphs of random Cayley graphs generated by a finite abelian group. [ABSTRACT FROM AUTHOR]

introduce the spectral decomposition property for measures and prove that a homeomorphism has the spectral decomposition property if and only if every Borel probability measure has the property too. Furthermore, we show that all shadowable measures for expansive homeomorphisms have the spectral decomposition property. Additionally, we provide illustrative examples relevant to these results. [ABSTRACT FROM AUTHOR]

Suppose V ν (·) is the pseudo-variance function of the Cauchy–Stieltjes Kernel (CSK) family K - (ν) generated by a non degenerate probability measure ν on the positive real line. Denote by Φ (ν) the law of large numbers for free multiplicative convolution given in [17]. An explicit expression of Φ (ν) is given in [14] in terms of the pseudo-variance function V ν (.) . In this paper, we give explicitly in terms of the pseudo-variance function V ν (.) the limiting distributions Φ (ν ⊠ t) , Φ ((D 1 / s (ν ⊞ s)) ⊠ t) , Φ ((D 1 / r (ν ⊎ r)) ⊠ t) and Φ ((B r (ν)) ⊠ t) for s > 1 , t > 1 and r > 0 , where B r (ν) = (ν ⊞ 1 + r) ⊎ 1 1 + r and D c (ν) denotes the dilation of measure ν by a number c ≠ 0 . Some examples of calculations of Φ (ν ⊠ t) , Φ ((D 1 / s (ν ⊞ s)) ⊠ t) , Φ ((D 1 / r (ν ⊎ r)) ⊠ t) and Φ ((B r (ν)) ⊠ t) are given for probability measures ν of importance in noncommutative probability. [ABSTRACT FROM AUTHOR]

We show global existence of a measure-valued solution to the Cauchy problem for the rectilinear Euler-Poisson equations, which govern flow of pressureless dust experiencing linear damping in the gravitational field generated by its own mass, and a given background charge field modeling exterior gravitational force. The velocity of the dust is a Borel measurable function, and the distribution of mass is given by time-dependent probability measures on the real line. Upon adapting the method established in Hynd (2020) [25] , we obtain the solution by construction of a probability measure on the path space, through approximation of finite sticky particles. [ABSTRACT FROM AUTHOR]

Let G be a locally compact hypergroup and let K be a compact subhypergroup of G. (G,K) is a Gelfand pair if Mc(G//K), the algebra of measures with compact support on the double coset G//K, is commutative for the convolution. In this paper, assuming that (G,K) is a Gelfand pair, we define and study the inverse Fourier transform on G and then establish a Bochner theorem for the pair (G,K). [ABSTRACT FROM AUTHOR]

In this article, we provide some new limiting laws related to the free multiplicative law of large numbers and involving free and Boolean additive convolutions. Some examples of these limiting laws are presented within the framework of non-commutative probability theory. [ABSTRACT FROM AUTHOR]

In this paper we study the optimal control problems of general Mckean-Vlasov for backward doubly stochastic differential equations (BDSDEs), in which the coefficients depend on the state of the solution process as well as of its law. We establish a stochastic maximum principle on the hypothesis that the control field is convex. For example, an example of a control problem is offered and solved using the primary result. [ABSTRACT FROM AUTHOR]

Let G be a locally compact group with the left Haar measure m G and let A = a n , k n , k = 0 ∞ be a strongly regular matrix. We show that if μ is a power bounded measure on G, then there exists an idempotent measure θ μ such that w*- lim n → ∞ ∑ k = 0 ∞ a n , k μ k = θ μ. If μ is a probability measure on a compact group G, then w*- lim n → ∞ ∑ k = 0 ∞ a n , k μ k = m ¯ H , where H is the closed subgroup of G generated by supp μ and m ¯ H is the measure on G defined by m ¯ H E : = m H E ∩ H for every Borel subset E of G. [ABSTRACT FROM AUTHOR]

For the set I(X) of probability measures on a compact Hausdorff space X, we propose a new way to introduce the topology by using the open subsets of the space X. Then, among other things, we give a new proof that for a compact Hausdorff space X, the space I(X) is also a compact Hausdorff space. For a Tychonoff space X, we consider the topological space Iτ(X) of τ-smooth idempotent probability measures on X and show that the space Iτ(X) is Čech-complete if and only if the given space X is Čech-complete.

Uncertainty is an inherent characteristic of a decision-making process. Occasionally, historical data may be insufficient to accurately estimate the probability distribution suitable for an unknown variable. In these situations, we deal with fuzzy stochastic variables in solving a problem. As a result, decision-makers, particularly those in the military, are confronted with numerous issues. This article discusses the maximum network flow interdiction under fuzzy stochastic hybrid conditions. The capacity of arcs has been treated as a fuzzy stochastic variable in this problem. The primary objective of this paper is to propose a model to the decision-maker that can be used to manage unknown factors in the network. Since this topic is explored concurrently in a stochastic and fuzzy environment, it is impossible to solve it directly. Consequently, three probability-possibility, probability-necessity, and probability-credibility techniques are utilized to transform it into a deterministic state. Eventually, the proposed model's efficacy is demonstrated by presenting a numerical example. [ABSTRACT FROM AUTHOR]

Undoubtedly, the best tool to help military decision-makers and commanders choose the appropriate strategy is to formulate models that are as close to the real world as possible. This is possible when these problems are presented in conditions of uncertainty. In this research, a multi-period dynamic interdiction problem in fuzzy stochastic conditions is investigated. In general, interdiction problem, scarce resources are allocated to degrade the enemy, whose behavior is formatted by the network optimization problem. In this problem, the defense forces in the role of interdictor try to minimize the maximum flow during the period so that at each stage the interdictor and the enemy are fully aware of the performance of the other side. Edge capacities in this model are considered as fuzzy stochastic variables. To solve the proposed model, first, the fuzzy stochastic dynamic interdiction problem is transformed into the deterministic dynamic interdiction problem with the help of the concepts of probability measure, credibility measure and chance constraint programming. Then, by creating the crisp two-level problem created by duality, it is transformed into a single-level problem, and then it is solved by using the generalization of Banders decomposition algorithm. Finally, the validity of the problem is evaluated by providing a numerical sample.

In this article, we provide some new limiting laws related to the free multiplicative law of large numbers and involving free and Boolean additive convolutions. Some examples of these limiting laws are presented within the framework of non-commutative probability theory.

We initiate a systematic investigation of group actions on compact median algebras via the corresponding dynamics on their spaces of measures. We show that a probability measure which is invariant under a natural push forward operation must be a uniform measure on a cube and use this to show that every amenable group action on a locally convex compact median algebra fixes a sub-cube. [ABSTRACT FROM AUTHOR]

The space of all probability measures having positive density function on a connected compact smooth manifold M, denoted by P(M)$\mathcal {P}(M)$, carries the Fisher information metric G. We define the geometric mean of probability measures by the aid of which we investigate information geometry of P(M)$\mathcal {P}(M)$, equipped with G. We show that a geodesic segment joining arbitrary probability measures μ1 and μ2 is expressed by using the normalized geometric mean of its endpoints. As an application, we show that any two points of P(M)$\mathcal {P}(M)$ can be joined by a unique geodesic. Moreover, we prove that the function ℓ defined by ℓ(μ1,μ2):=2arccos∫Mp1p2dλ$\ell \!\big (\mu _1, \mu _2\big):=2\arccos \int \nolimits _M \sqrt {p_1p_2}\,d\lambda$, μi=piλ$\mu _i=p_i \lambda$, i=1,2$i=1,2$, gives the Riemannian distance function on P(M)$\mathcal {P}(M)$. It is shown that geodesics are all minimal. [ABSTRACT FROM AUTHOR]

In this paper, we study the behavior of some topological and cardinal properties of topological spaces under the influence of the -kernel of a topological space . It has been proved that the -kernel of a topological space preserves the density and the network -weight of normal spaces. [ABSTRACT FROM AUTHOR]

Optimal allocation of the reliability improvement target is essential for the system optimization design. In order to solve the problems that the optimization model is with loss of generality and the validity of the optimal solution is weakened, an optimal allocation method is proposed by considering multiple correlation failures and risk uncertainty in this paper. Two new concepts are presented, such as independent failure results in basic risk, and correlation failure leads to disturbance risk. A risk assessment machinery of “actual risk = basic risk + disturbance risk” is proposed. The action mechanisms of the three correlation failures are studied based on the cooperation game theory, and the generalized risk models are given under probability measure. Considering the improvement cost, the expectation and the variance of the reduction of system risk, a multi-objective optimal allocation model is developed, which is solved by using the PSO algorithm. Finally, the proposed optimal allocation is implemented at the 2-stage NGW planetary reducer, and the results show that it is more efficient and feasible for engineering practice. [ABSTRACT FROM AUTHOR]

For a probability measure of compact support μ on the set P n of all positive definite matrices and t ∈ (0 , 1 ] , let P t (μ) be the unique positive solution of X = ∫ P n X ♯ t Z d μ (Z). In this paper, we show that if μ and ν are probability measures on P n and P m , respectively, and Φ : M n × M m → M k is a unital positive bilinear map, then Φ (P t (μ) , P t (ν)) ≤ P t (λ) for all t ∈ [ − 1 , 1 ] ∖ { 0 } , where λ is the push-forward of the product measure μ × ν by the mapping Φ | P n × P m . In particular, P t (μ) ⊗ P t (ν) ≤ P t (λ). Moreover, we show that P t (μ) ≤ I ⟹ P t p (ν) ≤ P t (μ) , for every p ≥ 1 , where ν (Z) = μ (Z 1 / p) and I denotes the identity matrix. [ABSTRACT FROM AUTHOR]

The novel concept of focality is introduced for Borel probability measures on compact Hausdorff topological spaces. We characterize focal Borel probability measures as those Borel probability measures that are strictly positive on every nonempty open subset. We also prove the existence of focal Borel probability measures on compact metric spaces. Lastly, we prove that the set of focal (regular) Borel probability measures is convex but not extremal in the set of all (regular) Borel probability measures. [ABSTRACT FROM AUTHOR]

In this paper, we establish necessary conditions of optimality for partially observed optimal control problems of Mckean–Vlasov type. The system is described by a controlled stochastic differential equation governed by Poisson random measure and an independent Brownian motion. The coefficients of the McKean–Vlasov system depend on the state of the solution process as well as of its probability law and the control variable. The proof of our result is based on Girsanov's theorem, variational equations and derivatives with respect to probability measure under convexity assumption. At the end of this paper, we apply our stochastic maximum principle to study partially observed linear quadratic control problem of McKean–Vlasov type with jumps and derive the explicit expression of the optimal control. [ABSTRACT FROM AUTHOR]

In this work, the separability, locally separability, weakly separability and locally weakly separability of the space of probability measures of an infinite compact space are studied. It is proved that an infinite compact space is a separable (locally separable), if and only if the space is a separable (locally separable). It was also proved that, if a space is a weakly separable (locally weakly separable), then the space is a weakly separable (locally weakly separable). [ABSTRACT FROM AUTHOR]

The quantization dimension of a probability measure on a metric compactum does not exceed the box dimension of the support of the measure. We prove the following intermediate value theorem for the upper quantization dimension: If is a metric compact space whose upper box dimension is equal to then for every real such that there exists a probability measure on whose support is and whose upper quantization dimension is . [ABSTRACT FROM AUTHOR]

Based on the Scarsini axioms characterizing concordance measures between two random variables X and Y linked by a copula C , we introduce and discuss convex concordance measures. In their simplest form, they are dependent, except for a few degenerated cases, on a value of C at a single point (x , y) such that 0 < y ≤ x ≤ 1 2 , and on the values at seven other points corresponding to this point, namely, (y , x) , (x , 1 − y) , (1 − y , x) , (1 − x , y) , (y , 1 − y) , (1 − x , 1 − y) and (1 − y , 1 − x). Our approach covers, among others, such classical concordance measures as Spearman's rho, Gini's gamma, Blomqvist's beta, but also the concordance measures introduced by Fuchs and Schmidt. Though some of the introduced constructions are related to the results of Edwards, Mikusiǹski and Taylor, they offer a genuine motivation and provide a more transparent view. [ABSTRACT FROM AUTHOR]

In the paper a set of idempotent probability measures and its maxplus-polar are described with a polar of the set of probability measures. An equality connecting a gauge of the set of probability measures and a max-plus-gauge of the set of idempotent probability measures is established. It is well known that every map between sets induces a map as between the corresponding sets of probability measures, as between the corresponding sets of idempotent probability measures. It is shown a relation between these induced maps. [ABSTRACT FROM AUTHOR]

Let be a locally compact Abelian group with dual group , let be a power bounded measure on , and let be a strongly regular matrix. We show that the sequence converges in the -norm for every if and only if is clopen in , where is the Fourier–Stieltjes transform of . If is a probability measure, then is clopen in if and only if the closed subgroup generated by the support of is compact. [ABSTRACT FROM AUTHOR]

For a continuous self-map f on a compact metric space X, we provide two simple examples: the first confirms that shadowing of (X, f) is not inherited by (M (X) , f ^) in general, and the other satisfies that both (X, f) and (M (X) , f ^) have no Li–Yorke pair, where M (X) be the space of all Borel probability measures on X. Then we prove that chain transitivity of (X, f) implies chain mixing of (M (X) , f ^) , and provide an example to deny the converse. For a non-autonomous system (X , f 0 , ∞) , we prove that weak mixing of (M (X) , f ^ 0 , ∞) implies that of (X , f 0 , ∞) , and give an example to deny the converse, where f 0 , ∞ = { f n } n = 0 ∞ is a sequence of continuous self-maps on X. We also prove that if f n is surjective for all n ≥ 0 , then chain mixing of (M (X) , f ^ 0 , ∞) always holds true, and shadowing of (M (X) , f ^ 0 , ∞) implies mixing of (X , f 0 , ∞) . If X = I is an interval, we obtain a sharp condition such that transitivity is equivalent between (I, f) and (M (I) , f ^) . Although (M (I) , f ^) has infinite topological entropy for any transitive system (I, f), we give an example such that (I , f 0 , ∞) is transitive but (M (I) , f ^ 0 , ∞) has zero topological entropy. [ABSTRACT FROM AUTHOR]

In this paper, a confidence set-membership state estimator is proposed for a class of polytopic linear parameter varying (LPV) systems with inexact scheduling variables. The set-bounded and Gaussian uncertainties are considered simultaneously in the process disturbances and measurement noises. The purpose of the proposed estimator is to achieve a confidence set of the state with given confidence level. Based on the polytopic LPV uncertain enclosure model, the set-bounded/Gaussian uncertainties of the state are given by using the worst case strategy. The size of the confidence set is minimized to get the optimal gain for the estimator. Meanwhile, the constrained zonotope is adopted to represent set-bounded uncertainties for more accurate results. Finally, a vehicle example is given to illustrate the effectiveness of proposed methods. • A confidence set-membership state estimator is designed for LPV systems. • The size of the confidence state set is minimized. • The LPV mapping of set-bounded uncertainties is calculated accurately with constrained zonotope. • Discussed the relationship between the confidence set-membership and other approaches. [ABSTRACT FROM AUTHOR]

t has been shown that if μ is a compactly supported probability measure on Mn+, then for every unit vector η∈ℂn, there exists a compactly supported probability measure (denoted by on ℝ+ so that the inequality ≤ Pt() (t∈(0,1]) holds. In particular, we consider a reverse of the above inequality and present some Pólya--Szegö type inequalities for power means of probability measures on positive matrices.

The novel concept of focality is introduced for Borel probability measures on compact Hausdorff topological spaces. We characterize focal Borel probability measures as those Borel probability measures that are strictly positive on every nonempty open subset. We also prove the existence of focal Borel probability measures on compact metric spaces. Lastly, we prove that the set of focal (regular) Borel probability measures is convex but not extremal in the set of all (regular) Borel probability measures.

Monte Carlo simulation (MCS) is useful for verifying analytical derivations and studying complex systems on an empirical basis. Although MCS is straightforward and does not require a priori knowledge of the sampled probability measures (PMs), it consumes a tremendous amount of computational resource and suffers from slow simulation procedures for sophisticated systems. To mitigate this drawback of MCS, we make full use of the monotone property and the logarithmic presentation convention of PMs in regard to power related metrics (PRMs) in communications science and propose an easy-to-implement interrupt mechanism to accelerate MCS for estimating PMs. To facilitate the programming on different platforms and provide a solid theoretical foundation, we present a generic implementation framework suited for estimating PMs with certain properties by MCS and analyze the underlying theory of the interrupt mechanism. In particular, we apply the de Moivre-Laplace theorem and analytic continuation to prove the asymptotic consistency under the indirect setup of MCS. We also design a hypothesis test for studying the proposed mechanism on a statistical basis. [ABSTRACT FROM AUTHOR]

The paper studies partially observed optimal control problems of general McKean–Vlasov differential equations, in which the coefficients depend on the state of the solution process as well as of its law and the control variable. By applying Girsanov's theorem with a standard variational technique, we establish a stochastic maximum principle on the assumption that the control domain is convex. As an application, partially observed linear-quadratic control problem is discussed. [ABSTRACT FROM AUTHOR]

The present paper aims at developing a theory of logical entropy for p -preserving systems (L, p, ψ), where p is a Bayesian gs -map on a bounded lattice L and ψ is a p -preserving lattice homomorphism. Having proved several desirable properties, we construct a Rokhlin-type metric involving logical conditional entropy of p -partitions of a Bayesian gs -space (L,p), and use it to establish a logical entropy version of Kolmogorov–Sinai theorem in the context. A formal mathematical foundation for the corresponding information theory is also presented in the end. [ABSTRACT FROM AUTHOR]

The goal of this paper is to extend the existence result of measure-valued solution to the Boltzmann equation in elastic interaction, given by Morimoto–Wang–Yang in (J Stat Phys 165:866–906, 2016), to the inelastic Boltzmann equation with moderately soft potentials, which is also an extensive work of our preceding result in the inelastic Maxwellian molecules case. We first prove the existence and uniqueness of measure-valued solution under Grad’s angular cutoff assumption, based on which, we further obtain the existence of non-cutoff solution, for both finite and infinite energy initial datum, by a delicate compactness argument. In addition, the moments propagation and energy dissipation properties are justified for the obtained measure-valued solution as well. [ABSTRACT FROM AUTHOR]

The article deals with the development of a methodological apparatus for quantifying the degree of economic risk in modeling uncertainty with the usу of fuzzy set theory. The study implements the task of further development of a fuzzy set method for risk measurement based on a probabilistic analogy for a case of simultaneous fuzziness of estimates of the criterion economic indicator (criterion) and its standard, which allow for a presentation form that is interval in terms of levels of membership ( levels). As part of this, an appropriate calculation model is formulated. Furthermore, the work pays attention to the situation when both the criterion and its standard are modeled by fuzzy values with a high degree of arbitrariness of their structure, which does not allow an interval, in terms of levels of membership, way of representing them. Within the methodological approach, based on the probabilistic analogy, a possible computational design is proposed for this purpose. Along with the above mentioned, the study distinguishes as practically significant a combined situation, when the structure for fuzzy assessment of the criterion economic indicator is characterized with a high degree of randomness, while the assessment of its standard assumes a presentation form that is interval in terms of membership level. Moreover, possible calculation model is formulated for the above statement of the problem under study. As a promising area for further scientific research on the problem field considered in the article, there identified studying and developing tools to support decision-making in management of economic systems that allow handling simultaneously different types of uncertainty.