On rough convergence of complex uncertain sequences

Complex uncertain variables are measurable functions from an uncertainty space to the set of complex numbers and are used to model complex uncertain quantities. The main purpose of this paper is to introduce rough convergence of complex uncertain sequences and study some convergence concepts namely rough convergence in measure, rough convergence in mean, rough convergence in distribution of complex uncertain sequences. Lastly some relationship between them have been investigated.

Convergence in distribution for randomly stopped random fields

Let X \mathbb {X} and Y \mathbb {Y} be two complete, separable, metric spaces, ξ ε ( x ) , x ∈ X \xi _\varepsilon (x), x \in \mathbb {X} and ν ε \nu _\varepsilon be, for every ε ∈ [ 0 , 1 ] \varepsilon \in [0, 1] , respectively, a random field taking values in space Y \mathbb {Y} and a random variable taking values in space X \mathbb {X} . We present general conditions for convergence in distribution for random variables ξ ε ( ν ε ) \xi _\varepsilon (\nu _\varepsilon ) that is the conditions insuring holding of relation, ξ ε ( ν ε ) ⟶ d ξ 0 ( ν 0 ) \xi _\varepsilon (\nu _\varepsilon ) \stackrel {\mathsf {d}}{\longrightarrow } \xi _0(\nu _0) as ε → 0 \varepsilon \to 0 .

Nonstandard Hulls of C*-Algebras and Their Applications

For the sake of providing insight into the use of nonstandard techniques à la A. Robinson and into Luxemburg’s nonstandard hull construction, we first present nonstandard proofs of some known results about C*-algebras. Then we introduce extensions of the nonstandard hull construction to noncommutative probability spaces and noncommutative stochastic processes. In the framework of internal noncommutative probability spaces, we investigate properties like freeness and convergence in distribution and their preservation by the nonstandard hull construction. We obtain a nonstandard characterization of the freeness property. Eventually we provide a nonstandard characterization of the property of equivalence for a suitable class of noncommutative stochastic processes and we study the behaviour of the latter property with respect to the nonstandard hull construction.

On Smooth Mesoscopic Linear Statistics of the Eigenvalues of Random Permutation Matrices

We study the limiting behavior of smooth linear statistics of the spectrum of random permutation matrices in the mesoscopic regime, when the permutation follows one of the Ewens measures on the symmetric group. If we apply a smooth enough test function f to all the determinations of the eigenangles of the permutations, we get a convergence in distribution when the order of the permutation tends to infinity. Two distinct kinds of limit appear: if $$f(0)\ne 0$$ f ( 0 ) ≠ 0 , we have a central limit theorem with a logarithmic variance; and if $$f(0) = 0$$ f ( 0 ) = 0 , the convergence holds without normalization and the limit involves a scale-invariant Poisson point process.

The $k$-Cut Model in Deterministic and Random Trees

The \(k\)-cut number of rooted graphs was introduced by Cai et al. as a generalization of the classical cutting model by Meir and Moon. In this paper, we show that all moments of the \(k\)-cut number of conditioned Galton-Watson trees converge after proper rescaling, which implies convergence in distribution to the same limit law regardless of the offspring distribution of the trees. This extends the result of Janson. Using the same method, we also show that the \(k\)-cut number of various random or deterministic trees of logarithmic height converges in probability to a constant after rescaling, such as random split-trees, uniform random recursive trees, and scale-free random trees.

Convergence of Complex Uncertain Double Sequences

Complex uncertain variables are measurable functions from an uncertainty space to the set of complex numbers and are used to model complex uncertain quantities. This paper introduces the convergence concepts of convergence almost surely (a.s.), convergence in measure, convergence in mean, convergence in distribution and convergence uniformly almost surely complex uncertain double sequences. In addition, relationships among the introduced classes of sequences have been introduced.

On rough convergence variables of triple sequences

Triple sequence convergence plays an extremely important role in the fundamental theory of mathematics. This paper contains four types of convergence concepts, namely, convergence almost surely, convergence incredibility, trust convergence in mean, and convergence in distribution, and discuss the relationship among them and some mathematical properties of those new convergence.