Vague Convergence Research Articles

Short cycles of random permutations with cycle weights: Point processes approach

We study the asymptotic behavior of short cycles of random permutations with cycle weights. More specifically, on a specially constructed metric space whose elements encode all possible cycles, we consider a point process containing all information on cycles of a given random permutation on {1,…,n}. The main result of the paper is the distributional convergence with respect to the vague topology of the above processes towards a Poisson point process as n→∞ for a wide range of cycle weights. As an application, we give several limit theorems for various statistics of cycles.

On the asymptotic spectral distribution of increasing size matrices: test functions, spectral clustering, and asymptotic estimates of outliers

Let {An}n be a sequence of square matrices such that size(An)=dn→∞ as n→∞. We say that {An}n has an asymptotic spectral distribution described by a Lebesgue measurable function f:D⊂Rk→C if, for every continuous function F:C→C with bounded support,limn→∞⁡1dn∑i=1dnF(λi(An))=1μk(D)∫DF(f(x))dx, where μk is the Lebesgue measure in Rk and λ1(An),…,λdn(An) are the eigenvalues of An. In the last decades, the asymptotic spectral distribution of increasing size matrices has become a subject of investigation by several authors. Special attention has been devoted to matrices An arising from the discretization of differential equations, whose size dn diverges to ∞ along with the mesh-fineness parameter n. However, despite the popularity that the topic has reached nowadays, the role of the so-called test functions F appearing in the above limit relation has been inexplicably neglected so far. In particular, a natural question such as “Which is the largest set of test functions F for which the above limit relation is satisfied?” is still unanswered. In the present paper, we provide a definitive answer to this question by identifying the largest set of test functions F for which the above limit relation is satisfied. We also present some applications of this result to the analysis of spectral clustering, including new asymptotic estimates on the number of outliers. A special attention is devoted to the so-called “inner outliers” of Hermitian block Toeplitz matrices. We conclude the paper with an interpretation of the main result in the context of the vague convergence of probability measures.

Effective weak and vague convergence of measures on the real line

We expand our effective framework for weak convergence of measures on the real line by showing that effective convergence in the Prokhorov metric is equivalent to effective weak convergence. In addition, we establish a framework for the study of the effective theory of vague convergence of measures. We introduce a uniform notion and a non-uniform notion of vague convergence, and we show that both these notions are equivalent. However, limits under effective vague convergence may not be computable even when they are finite. We give an example of a finite incomputable effective vague limit measure, and we provide a necessary and sufficient condition so that effective vague convergence produces a computable limit. Finally, we determine a sufficient condition for which effective weak and vague convergence of measures coincide. As a corollary, we obtain an effective version of the equivalence between classical weak and vague convergence of sequences of probability measures.

Minimum Riesz energy problems with external fields

The paper deals with minimum energy problems in the presence of external fields with respect to the Riesz kernels |x−y|α−n, 0<α<n, on Rn, n⩾2. For quite a general (not necessarily lower semicontinuous) external field f, we obtain necessary and/or sufficient conditions for the existence of λA,f minimizing the Gauss functional∫|x−y|α−nd(μ⊗μ)(x,y)+2∫fdμ over all positive Radon measures μ with μ(Rn)=1, concentrated on quite a general (not necessarily closed) A⊂Rn. We also provide various alternative characterizations of the minimizer λA,f, analyze the continuity of both λA,f and the modified Robin constant for monotone families of sets, and give a description of the support of λA,f. The significant improvement of the theory in question thereby achieved is due to a new approach based on the close interaction between the strong and the vague topologies, as well as on the theory of inner balayage, developed recently by the author.

Limit theory for U-statistics under geometric and topological constraints with rare events

AbstractWe study the geometric and topological features of U-statistics of order k when the k-tuples satisfying geometric and topological constraints do not occur frequently. Using appropriate scaling, we establish the convergence of U-statistics in vague topology, while the structure of a non-degenerate limit measure is also revealed. Our general result shows various limit theorems for geometric and topological statistics, including persistent Betti numbers of Čech complexes, the volume of simplices, a functional of the Morse critical points, and values of the min-type distance function. The required vague convergence can be obtained as a result of the limit theorem for point processes induced by U-statistics. The latter convergence particularly occurs in the $\mathcal M_0$ -topology.

A New Approach to Vague Soft Bi-Topological Spaces

Fuzzy soft topology considers only membership value. It has nothing to do with the non-membership value. So an extension was needed in this direction. Vague soft topology addresses both membership and non-membership values simultaneously. Sometimes vague soft topology (single structure) is unable to address some complex structures. So an extension to vague soft bi-topology (double structure) was needed in this direction. To make this situation more meaningful, a new concept of vague soft bi-topological space is introduced and its structural characteristics are attempted with a new definition. In this article, new concept of vague soft bi-topological space (VSBTS) is initiated and its structural behaviors are attempted. This approach is based on generalized vague soft open sets, known as vague soft open sets. An ample of examples are given to understand the structures. For the non-validity of some results, counter examples are provided to pay the price. Pair-wise vague soft open and pair-wise vague soft close sets are also addressed with examples in (VSBTS). Vague soft separation axioms are initiated in (VSBTS) concerning soft points of the space. Other separation axioms are also addressed relative to soft points of the space. Vague soft bi-topological properties are studied with the application of vague soft open sets with respect to soft points of the spaces. The characterization of vague soft close as well as vague soft open sets, characteristics of Bolzano Weirstrass property, vague soft compactness and its marriage with sequences, interconnection between sequential compactness and countable compactness in (VSBTS) with respect to soft open sets are addressed.

An Absolutely New Attempt to Vague Soft Bitopological Spaces Basis at Newly Defined Operations regarding Vague Soft Points

In this study, new operations of union, intersection, and complement are defined with the help of vague soft sets in a new way that is in both true and false statements, union is defined with maximum, and intersection is defined with minimum. On the basis of these operations, vague soft topology is defined. Pairwise vague soft open sets and pairwise vague soft closed sets are defined in vague soft bitopological structures (VSBTS). Moreover, generalized vague soft open sets are introduced in VSBTS concerning soft points of the space. On the basis of generalized vague soft open sets, separation axioms are also introduced. In continuation, these separations axioms are engaged with other important results in VSBTS.

The space of invariant measures for countable Markov shifts

It is well known that the space of invariant probability measures for transitive sub-shifts of finite type is a Poulsen simplex. In this article we prove that in the non-compact setting, for a large family of transitive countable Markov shifts, the space of invariant sub-probability measures is a Poulsen simplex and that its extreme points are the ergodic invariant probability measures together with the zero measure. In particular, we obtain that the space of invariant probability measures is a Poulsen simplex minus a vertex and the corresponding convex combinations. Our results apply to finite entropy non-locally compact transitive countable Markov shifts and to every locally compact transitive countable Markov shift. In order to prove these results we introduce a topology on the space of measures that generalizes the vague topology to a class of non-locally compact spaces, the topology of convergence on cylinders. We also prove analogous results for suspension flows defined over countable Markov shifts.

On the Existence of Global Smooth Solutions to the Parabolic–Elliptic Keller–Segel System with Irregular Initial Data

We consider the parabolic–elliptic Keller–Segel system *ut=Δu-χ∇·(u∇v),0=Δv-v+u\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{aligned} u_t&= \\Delta u - \\chi \ abla \\cdot (u \ abla v), \\\\ 0&= \\Delta v - v + u \\end{aligned} \\right. \\end{aligned}$$\\end{document}in a smooth bounded domain Omega subseteq {mathbb {R}}^n, nin {mathbb {N}}, with Neumann boundary conditions. We look at both chemotactic attraction (chi > 0) and repulsion (chi < 0) scenarios in two and three dimensions. The key feature of interest for the purposes of this paper is under which conditions said system still admits global classical solutions due to the smoothing properties of the Laplacian even if the initial data is very irregular. Regarding this, we show for initial data mu in {mathcal {M}}_+({overline{Omega }}) that, if eithern = 2, chi < 0 orn = 2, chi > 0 and the initial mass is small orn = 3, chi < 0 and mu = f in L^p(Omega ), p > 1 holds, it is still possible to construct global classical solutions to (star ), which are continuous in t = 0 in the vague topology on {mathcal {M}}_+({overline{Omega }}).

On the (dis)continuity of the Fourier transform of measures

In this paper, we will study the continuity of the Fourier transform of measures with respect to the vague topology. We show that the Fourier transform is vaguely discontinuous on R, but becomes continuous when restricting to a class of Fourier transformable measures such that either the measures, or their Fourier transforms are equi-translation bounded. We discuss continuity of the Fourier transform in the product and norm topology. We show that vague convergence of positive definite measures implies the equi translation boundedness of the Fourier transforms, which explains the continuity of the Fourier transform on the cone of positive definite measures. In the appendix, we characterize vague precompactness of a set of measures in arbitrary LCAG, and the necessity of second countability property of a group for defining the autocorrelation measure.

A Markov process for an infinite interacting particle system in the continuum

An infinite system of point particles placed in $\mathds{R}^d$ is studied. Its constituents perform random jumps with mutual repulsion described by a translation-invariant jump kernel and interaction potential, respectively. The pure states of the system are locally finite subsets of $\mathds{R}^d$, which can also be interpreted as locally finite Radon measures. The set of all such measures $\Gamma$ is equipped with the vague topology and the corresponding Borel $\sigma$-field. For a special class $\mathcal{P}_{\rm exp}$ of (sub-Poissonian) probability measures on $\Gamma$, we prove the existence of a unique family $\{P_{t,\mu}: t\geq 0, \ \mu \in \mathcal{P}_{\rm exp}\}$ of probability measures on the space of cadlag paths with values in $\Gamma$ that solves a restricted initial-value martingale problem for the mentioned system. Thereby, a Markov process with cadlag paths is specified which describes the stochastic dynamics of this particle system.

A concept of weak Riesz energy with application to condensers with touching plates

We proceed further with the study of minimum weak Riesz energy problems for condensers with touching plates, initiated jointly with Fuglede (Potential Anal 51:197–217, 2019). Having now added to the analysis constraint and external source of energy, we obtain a Gauss type problem, but with weak energy involved. We establish sufficient and/or necessary conditions for the existence of solutions to the problem and describe their potentials. Treating the solution as a function of the condenser and the constraint, we prove its continuity relative to the vague topology and the topologies determined by the weak and standard energy norms. We show that the criteria for the solvability thus obtained fail in general once the problem is reformulated in the setting of standard energy, thereby justifying an advantage of weak energy when dealing with condensers with touching plates.

On the Zeros of Non-Analytic Random Periodic Signals

Abstract In this paper, we investigate the local universality of the number of zeros of a random periodic signal of the form $S_n(t)=\sum _{k=1}^n a_k f(k t)$, where $f$ is a $2\pi -$periodic function satisfying weak regularity conditions and where the coefficients $a_k$ are i.i.d. random variables, which are centered with unit variance. In particular, our results hold for continuous piecewise linear functions. We prove that the number of zeros of $S_n(t)$ in a shrinking interval of size $1/n$ converges in law as $n$ goes to infinity to the number of zeros of a Gaussian process whose explicit covariance only depends on the function $f$ and not on the common law of the random coefficients $(a_k)$. As a byproduct, this entails that the point measure of the zeros of $S_n(t)$ converges in law to an explicit limit on the space of locally finite point measures on $\mathbb R$ endowed with the vague topology. The standard tools involving the regularity or even the analyticity of $f$ to establish such kind of universality results are here replaced by some high-dimensional Berry–Esseen bounds recently obtained in [ 7]. The latter allow us to prove functional Central Limit Theorems in $C^1$ or Lipschitz topology in situations where usual criteria cannot be applied due to the lack of regularity.

Constrained minimum Riesz energy problems for a condenser with intersecting plates

We study the constrained minimum energy problem with an external field relative to the α-Riesz kernel x−yα−n of order α ∈ (0, n) for a generalized condenser A = (Ai)i∈I in ℝn, n ⩾ 3, whose oppositely charged plates intersect each other over a set of zero capacity. Conditions sufficient for the existence of minimizers are found, and their uniqueness and vague compactness are studied. Conditions obtained are shown to be sharp. We also analyze continuity of the minimizers in the vague and strong topologies when the condenser and the constraint both vary, describe the weighted equilibrium vector potentials, and single out their characteristic properties. Our arguments are based particularly on the simultaneous use of the vague topology and a suitable semimetric structure on a set of vector measures associated with A, and the establishment of completeness theorems for proper semimetric spaces. The results remain valid for the logarithmic kernel on ℝ2 and A with compact Ai, i ∈ I. The study is illustrated by several examples.

Convergence to scale-invariant Poisson processes and applications in Dickman approximation

We study weak convergence of a sequence of point processes to a scale-invariant simple point process. For a deterministic sequence $(z_{n})_{n\in \mathbb {N}}$ of positive real numbers increasing to infinity as $n \to \infty $ and a sequence $(X_{k})_{k\in \mathbb {N}}$ of independent non-negative integer-valued random variables, we consider the sequence of point processes \[ \nu _{n}=\sum _{k=1}^{\infty }X_{k} \delta _{z_{k}/z_{n}}, \quad n \in \mathbb {N}, \] and prove that, under some general conditions, it converges vaguely in distribution to a scale-invariant Poisson process $\eta _{c}$ on $(0,\infty )$ with the intensity measure having the density $ct^{-1}$, $t\in (0,\infty )$. An important motivating example from probabilistic number theory relies on choosing $X_{k} \sim {\mathrm {Geom}}(1-1/p_{k})$ and $z_{k}=\log p_{k}$, $k \in \mathbb {N}$, where $(p_{k})_{k \in \mathbb {N}}$ is an enumeration of the primes in increasing order. We derive a general result on convergence of the integrals $\int _{0}^{1} t \nu _{n}(dt)$ to the integral $\int _{0}^{1} t \eta _{c}(dt)$, the latter having a generalized Dickman distribution, thus providing a new way of proving Dickman convergence results. We extend our results to the multivariate setting and provide sufficient conditions for vague convergence in distribution for a broad class of sequences of point processes obtained by mapping the points from $(0,\infty )$ to $\mathbb {R}^{d}$ via multiplication by i.i.d. random vectors. In addition, we introduce a new class of multivariate Dickman distributions which naturally extends the univariate setting.

Aspect-Oriented Modeling of Spatial Data Interpolation for Estimating Missing Data in Internet of Things (IoT) Service Discovery

Abstract The Internet of Things (IoT) is a model of future Internet and pervasive computing that has its own particular difficulties gained from the Internet as far as adaptability, vague topology, and so on are concerned. The proposed work means to determine the difficulties posed by IoT in the service discovery field. Most of the data analytics algorithms applied for data collected through sensors and actuators assume that the data are complete such that each property of the instances is filled with the appropriate value. These data have temporal and spatial correlation between them, and missing such data results in a significant decrease in accuracy and reliability of data analysis performed. Considering the importance of estimating the spatial data and the intricacies involved in estimating it using interpolation techniques, the proposed work bases its system development using an aspect-oriented programming improvement technique, thereby addressing the interpolation strategy as a cross-cutting aspect and reducing the complexity involved thereof. The proposal analyzes the situation of missing data and appropriately weaves the aspect and the application together, thereby decreasing the complexity in handling the interpolating data. The woven aspect estimates the missing data using an inverse distance weighting method and updates the information. Observation of the experimental results reveals significant improvement in response time compared with estimating the unknown value in a conventional manner.

Constrained minimum Riesz and Green energy problems for vector measures associated with a generalized condenser

For a finite collection $\mathbf{A}=(A_i)_{i \in I}$ of locally closed sets in $\mathbb R ^n$, $n \geqslant 3$, with the sign $s_i=\pm 1$ prescribed such that the oppositely charged plates are mutually disjoint, we consider the minimum energy problem relative to the $\alpha$-Riesz kernel $|x − y|^{\alpha-n}, \alpha \in (0,2]$, over positive vector Radon measures $\mathbf{μ}=(\mu^i)_{i \in I}$ such that each $\mu^i, i \in I$, is carried by $A_i$ and normalized by $\mu^1(A_i)=a_i \in (0, \infty)$, while the interaction between $\mu^i, i \in I$ , is determined by the matrix $(s_is_j)_{i,j\in I^\bullet}$. We show that, though the closures of oppositely charged plates may intersect each other even in a set of nonzero capacity, this problem has a solution $\mathbf{λ}_\mathbf{A}^\xi=(\lambda_\mathbf{A}^i)_{i \in I}$ (also in the presence of an external field) if we restrict ourselves to $\mathbf{μ}$ with $\mu^i\leqslant\xi^i, i \in I$, where the constraint $\mathbf{ξ}=(\xi^i)_{i \in I}$ is properly chosen. We establish the sharpness of the suffcient conditions on the solvability thus obtained, provide descriptions of the weighted vector $\alpha$-Riesz potentials of the solutions, single out their characteristic properties, and analyze the supports of the $\lambda_\mathbf{A}^i, i \in I$. Our approach is based on the simultaneous use of the vague topology and an appropriate semimetric structure defined in terms of the $\alpha$-Riesz energy on a set of vector measures associated with $\mathbf{A}$, as well as on the establishment of an intimate relationship between the constrained minimum $\alpha$-Riesz energy problem and a constrained minimum $\alpha$-Green energy problem, suitably formulated. The results are illustrated by examples.

A note on vague convergence of measures

We propose a new approach to vague convergence of measures based on the general theory of boundedness due to Hu (1966). The article explains how this connects and unifies several frequently used types of vague convergence from the literature. Such an approach allows one to translate already developed results from one type of vague convergence to another. We further analyze the corresponding notion of vague topology and give a new and useful characterization of convergence in distribution of random measures in this topology.

Condensers with infinitely many touching Borel plates and minimum energy problems

Defining a condenser in a locally compact space as a locally finite, countable collection of Borel sets $A_i$, $i\in I$, with the sign $s_i=\pm1$ prescribed such that $A_i\cap A_j=\varnothing$ whenever $s_is_j=-1$, we consider a minimum energy problem with an external field over infinite dimensional vector measures $(\mu^i)_{i\in I}$, where $\mu^i$ is a suitably normalized positive Radon measure carried by $A_i$ and such that $\mu^i\leqslant\xi^i$ for all $i\in I_0$, $I_0\subset I$ and constraints $\xi^i$, $i\in I_0$, being given. If $I_0=\varnothing$, the problem reduces to the (unconstrained) Gauss variational problem, which is in general unsolvable even for a condenser of two closed, oppositely signed plates in $\mathbb R^3$ and the Coulomb kernel. Nevertheless, we provide sufficient conditions for the existence of solutions to the stated problem in its full generality, establish the vague compactness of the solutions, analyze their uniqueness, describe their weighted potentials, and single out their characteristic properties. The strong and the vague convergence of minimizing nets to the minimizers is also studied. The phenomena of non-existence and non-uniqueness of solutions to the problem are illustrated by examples. The results obtained are new even for the classical kernels on $\mathbb R^n$, $n\geqslant2$, and closed $A_i$, $i\in I$, which is important for applications.

Equilibria Configurations for Epitaxial Crystal Growth with Adatoms

The behavior of a surface energy $\mathcal F(E,u)$, where $E$ is a set of finite perimeter and $u\in L^1(\partial^* E, \mathbb R_+)$ is studied. These energies have been recently considered in the context of materials science to derive a new model in crystal growth that takes into account the effect of atoms freely diffusing on the surface (called adatoms), which are responsible for morphological evolution through an attachment and detachment process. Regular critical points, existence and uniqueness of minimizers are discussed and the relaxation of $\mathcal F$ in a general setting under the $L^1$ convergence of sets and the vague convergence of measures is characterized. This is part of an ongoing project aimed at an analytical study of diffuse interface approximations of the associated evolution equations.