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Bregman distance iterative methods for solving optimization problems are important and interesting because of the numerous applications of Bregman distance techniques. In this paper, for solving a split feasibility problem, we introduce a new Bregman projection algorithm and construct two selection strategies of stepsizes. Moreover, a relaxed Bregman projection algorithm is proposed with two selection strategies of stepsizes, where the two closed and convex sets are both level sets of convex functions. Weak convergence results of the proposed algorithms are obtained under suitable assumptions. In addition, using the proposed algorithms with different Bregman distances, a numerical experiment solving signal processing problem is also given to demonstrate the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, weak convergence of balanced stochastic one-step methods and especially balanced stochastic Runge–Kutta (SRK) methods for Itô multidimensional stochastic differential equations is analyzed. Generalizing a corresponding result obtained by H. Schurz for the standard Euler method, it is shown that under certain conditions, balanced one-step methods preserve the weak convergence properties of their underlying methods. As an application, this allows to prove the weak convergence order of the balanced SRK methods presented in earlier work by A. Rathinasamy, P. Nair and D. Ahmadian. [ABSTRACT FROM AUTHOR]

In this paper, the split feasibility problem is studied in real Banach spaces. Through the W-mapping, a new iterative algorithm with the inertial technique for solving the split feasibility problem is proposed, which the step size is self-adaptive and no prior estimation of operator norm is required. We prove that the proposed algorithm converges weakly to a solution of the split feasibility problem under some mild conditions. Finally, the effectiveness of the proposed algorithm is indicated by numerical experiments. Our results are innovative and can enrich recently announced related results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of Bregman nonexpansive sequences and mappings in a Banach space, prove an ergodic theorem and a fixed point theorem, and some weak convergence theorems for such sequences and maps with rather mild assumptions on the Banach space. These results extend our previous results for ø-nonexpansive sequences and give a partial answer to the open problem raised in 1981 by Djafari Rouhani for nonexpansive sequences in a Banach space. [ABSTRACT FROM AUTHOR]

We introduce here a diffusion-type approximation of the ruin probability both in finite and infinite time for a two-dimensional risk process, where claims and premiums are shared with a predetermined proportion. This type of process is often called the insurer–reinsurer model. We assume that the flow of claims is governed by a general renewal process. A simple ruin probability formula for the model is known only in infinite time for the special case of the Poisson process and exponentially distributed claims. Therefore, there is a need for simple analytical approximations. In the literature, in the infinite-time case, for the Poisson process, a De Vylder-type approximation has already been introduced. The idea of the diffusion approximation presented here is based on the weak convergence of stochastic processes, which enables one to replace the original risk process with a Brownian motion with drift. By applying this idea to the insurer–reinsurer model, we obtain simple ruin probability approximations for both finite and infinite time. We check the usefulness of the approximations by studying several claim amount distributions and comparing the results with the De Vylder-type approximation and Monte Carlo simulations. All the results show that the proposed approximations are promising and often yield small relative errors. [ABSTRACT FROM AUTHOR]

We analyze multilayer neural networks in the asymptotic regime of simultaneously (a) large network sizes and (b) large numbers of stochastic gradient descent training iterations. We rigorously establish the limiting behavior of the multilayer neural network output. The limit procedure is valid for any number of hidden layers, and it naturally also describes the limiting behavior of the training loss. The ideas that we explore are to (a) take the limits of each hidden layer sequentially and (b) characterize the evolution of parameters in terms of their initialization. The limit satisfies a system of deterministic integro-differential equations. The proof uses methods from weak convergence and stochastic analysis. We show that, under suitable assumptions on the activation functions and the behavior for large times, the limit neural network recovers a global minimum (with zero loss for the objective function). [ABSTRACT FROM AUTHOR]

The class of balanced implicit methods (BIMs) has been introduced by Milstein, Platen and Schurz (1998) for numerical integration of d -dimensional systems of ordinary Itô-type stochastic differential equations (SDEs) driven by independent Wiener processes. They also established the rate r = 0.5 of mean square convergence towards exact solution along equidistant partitions under appropriate assumptions. The other major convergence criterion is that of weak convergence (i.e. the convergence of moments E [ f (X) ]) of them with respect to certain test classes of functions f and remains to be an open question. We fill that gap by proving weak convergence of a subclass of BIMs with rate r = 1.0 with respect to the test class of C 2 -functions and with variable step sizes. This class of stochastic-numerical approximations represents a linear-implicit family of schemes stabilizing p -th moments by a computationally efficient implementation. The novel proof relies on a random version of Clarkson-Beckner inequality to estimate its p -th moments. Moreover, we prove stability and boundedness of a more general class of these methods in any dimension d. [ABSTRACT FROM AUTHOR]

We provide scaling limits for the block counting process and the fixation line of Λ-coalescents as the initial state n tends to infinity under the assumption that the measure Λ on [0, 1] satisfies ∫[0,1] u-1|Λ - bλ|(du) < ∞ for some b ≥ 0. Here λ denotes the Lebesgue measure on [0, 1]. The main result states that the block counting process, properly transformed, converges in the Skorohod space to a generalized Ornstein-Uhlenbeck process as n tends to infinity. The result is applied to beta coalescents with parameters 1 and b > 0. We split the generators into two parts by additively decomposing Λ into a 'Bolthausen-Sznitman part' bλ and a 'dust part' Λ - bλ and then prove the uniform convergence of both parts separately. [ABSTRACT FROM AUTHOR]

In this paper, an iterative method with alternated inertial extrapolation step is proposed to solve the split feasibility problem. The stepsize in the proposed algorithm uses the self adaptive technique, which does not depend on the prior information of the operator norm. The weak convergence is proved under suitable conditions. Finally, a numerical example is given to illustrate the effectiveness of our algorithm. [ABSTRACT FROM AUTHOR]

This paper focuses on stability of stochastic hybrid systems with random delay driven by a singularly perturbed Markov chain. By weak convergence and the martingale method, the limit system is obtained. Using the limit system as a bridge, this paper establishes the moment exponential stability of the original system by Razumikhin-type techniques. Finally, an example is given to illustrate this result. [ABSTRACT FROM AUTHOR]

In this paper, we use the dual variable to propose a new iterative algorithm for solving the split common fixed-point problem of quasi-nonexpansive mappings in real Hilbert spaces. Under suitable conditions, we establish a weak convergence theorem of the proposed algorithm and obtain a related result for the split common fixed-point problem of firmly quasi-nonexpansive mappings. Some numerical experiments are given to illustrate the efficiency of the proposed iterative algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we study to approximate solutions of quasi-equilibrium problems in Banach spaces, which generalizes equilibrium problems and quasi-variational inequalities. In fact, we prove weak convergence of the sequence generated by the proximal point method to a solution of the quasi-equilibrium problem. Then we show that the generated sequence converges strongly to a solution of the quasi-equilibrium problem. [ABSTRACT FROM AUTHOR]

Nourdin and Peccati (2009a) established a neat characterization of Gamma approximation on a fixed Wiener chaos in terms of convergence of only the third and fourth cumulants. In this paper, we provide an optimal rate of convergence in the d2-distance in terms of the maximum of the third and fourth cumulants analogous to the result for normal approximation in Nourdin and Peccati (2015). In order to achieve our goal, we introduce a novel operator theory approach to Stein's method. The recent development in Stein's method for the Gamma distribution of Döbler and Peccati (2018) plays a pivotal role in our analysis. Several examples in the context of quadratic forms are considered to illustrate our optimal bound. [ABSTRACT FROM AUTHOR]

General properties of convergence in distribution for fuzzy random variables are studied as regards its interplay with the structure of the space of fuzzy sets. In particular, its behaviour with respect to taking tuples of fuzzy random variables, adding, multiplying by a scalar, taking the union, preserving inclusion ordering, and subsuming convergence in distribution of random sets is established. [ABSTRACT FROM AUTHOR]

It is shown that in general the popular coefficient alpha estimator for reliability of multi-component measuring instruments converges almost surely to a quantity that is not equal to the population reliability coefficient. This convergence with probability 1 is a stronger statement than convergence in probability (consistency) and convergence in distribution for the alpha estimator, which have been studied in the past. In the special case of congeneric measures with uncorrelated errors and equal loadings on the common true score, the alpha estimator converges almost surely to the population reliability coefficient that equals population alpha, which implies also its consistency as a reliability estimator. When the loadings are unequal but sufficiently high and similar, the alpha estimator converges almost surely to population alpha that is essentially indistinguishable from the population reliability coefficient, which implies alpha's approximate consistency then. For the general case, the results entail that the alpha estimator is not a consistent estimator of reliability. The findings add to the critical literature on coefficient alpha in the general case, as well as to the justification of its use as a dependable measuring instrument reliability estimator in special cases and settings resulting under appropriate restrictive conditions, and are illustrated using a numerical example. [ABSTRACT FROM AUTHOR]

Abstract A fundamental algorithm for selecting ranks from a finite subset of an ordered set is Radix Selection. This algorithm requires the data to be given as strings of symbols over an ordered alphabet, e.g., binary expansions of real numbers. Its complexity is measured by the number of symbols that have to be read. In this paper the model of independent data identically generated from a Markov chain is considered. The complexity is studied as a stochastic process indexed by the set of infinite strings over the given alphabet. The orders of mean and variance of the complexity and, after normalization, a limit theorem with a centered Gaussian process as limit are derived. This implies an analysis for two standard models for the ranks: uniformly chosen ranks, also called grand averages, and the worst case rank complexities which are of interest in computer science. For uniform data and the asymmetric Bernoulli model (i.e. memoryless sources), we also find weak convergence for the normalized process of complexities when indexed by the ranks while for more general Markov sources these processes are not tight under the standard normalizations. [ABSTRACT FROM AUTHOR]

We consider sequences of random variables living in a finite sum of Wiener chaoses. We find necessary and sufficient conditions for convergence in law to a target variable living in the sum of the first two Wiener chaoses. Our conditions hold notably for sequences of multiple Wiener integrals. Malliavin calculus and in particular the Γ-operators are used. Our results extend previous findings by Azmoodeh et al. (2015) and are applied to central and non-central convergence situations. Our methods are applied to investigate stable convergence and we exclude certain classes of random variables as target variables for sequences living in a fixed Wiener chaos. [ABSTRACT FROM AUTHOR]

We consider reaction-diffusion systems with random rapidly oscillating coefficient. We do not assume any Lipschitz condition for the nonlinear function in the system, so, the uniqueness theorem for the corresponding initial-value problem may not hold for the considered reaction-diffusion system. Under the assumption that the random function is ergodic and statistically homogeneous in space variables we prove that the trajectory attractors of these systems tend in a weak sense to the trajectory attractors of the homogenized reaction-diffusion systems whose coefficient is the average of the corresponding term of the original systems. [ABSTRACT FROM AUTHOR]

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Sufficient conditions for the weak convergence of the distributions of the random variables (1 - x)ξx as x → 1- to the limiting gamma-distribution are put forward. The random variable ξx has power-series distribution with radius of convergence 1 and parameter x ∈ (0, 1). Limit theorems for the probabilities P{ξx = k} are proposed. Asymptotic expansions of local probabilities are derived for sums of independent identically distributed variables with the same distribution as ξx in a triangular array with x → 1-. For the corresponding general allocation scheme, local limit theorems for the joint distributions of the occupancies of the cells are obtained. [ABSTRACT FROM AUTHOR]

In this paper we discuss weak convergence of continuous-time Markov chains to a non-symmetric pure jump process. We approach this problem using Dirichlet forms as well as semimartingales. As an application, we discuss how to approximate a given Markov process by Markov chains. [ABSTRACT FROM AUTHOR]

The replacement of indicator functions by integrated beta kernels in the definition of the empirical tail dependence function is shown to produce a smoothed version of the latter estimator with the same asymptotic distribution but superior finite-sample performance. The link of the new estimator with the empirical beta copula enables a simple but effective resampling scheme. [ABSTRACT FROM AUTHOR]

In this paper, we consider the averaging principle for one dimensional stochastic Burgers equation with slow and fast time-scales. Under some suitable conditions, we show that the slow component strongly converges to the solution of the corresponding averaged equation. Meanwhile, when there is no noise in the slow component equation, we also prove that the slow component weakly converges to the solution of the corresponding averaged equation with the order of convergence 1 − r , for any 0 < r < 1 . [ABSTRACT FROM AUTHOR]

In this paper, we present two penalty-splitting inspired iteration schemes (PFFSA) and (PFFSA) for hierarchical equilibrium problems in Hilbert space. Based on the Opial-Passty lemma, we propose weak ergodic convergence and weak convergence of the iterative sequences generated by the Forward-Forward algorithm (PFFSA) and the Forward-Backward algorithm (PFFSA), which are proved under quite mild conditions: the bifunction of the two level equilibrium problems are supposed pseudomonotone. For strong convergence, we first add a strong monotonicity condition on the objective bifunction. We present after, a strong convergence result of algorithm (PFFSA) by adding a topological assumption, i.e. the objective bifunction is of class . Some examples are given to illustrate our results. The first example deals with pseudomonotone variational inequalities and convex minimization problem in the upper level problem. In the second one, we propose a convex minimization in the lower-level problem, where strong convergence of (PFFSA) to a minimum point is insured under infcompactness condition for objective function. These convergence results are new and generalize some recent results in this field. [ABSTRACT FROM AUTHOR]

Employing the weak convergence method, based on a variational representation for expected values of positive functionals of a Brownian motion, we investigate moderate deviation for a class of stochastic differential delay equations with small noises, where the coefficients are allowed to be highly nonlinear growth with respect to the variables. Moreover, we obtain the central limit theorem for stochastic differential delay equations which the coefficients are polynomial growth with respect to the delay variables. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a contraction algorithm for solving monotone variational inclusion problem. To reach this goal, our main iterative algorithm combine Dong’s projection and contraction algorithm with resolvent operator. Under suitable assumptions, we prove that the sequence generated by our main iterative algorithm converges weakly to the solution of the considered problem. Finally, we give two numerical examples to verify the feasibility of our main algorithm. [ABSTRACT FROM AUTHOR]

We study the hydrodynamics of compressible flows of active liquid crystals in the Beris--Edwards hydrodynamics framework, using the Landau--de Gennes Q-tensor order parameter to describe liquid crystalline ordering. We prove the existence of global weak solutions for this active system in three space dimensions by the three-level approximations and weak convergence argument. New techniques and estimates are developed to overcome the difficulties caused by the active terms. [ABSTRACT FROM AUTHOR]

The present paper deals with new results in the study of a class of elliptic hemivariational inequalities in reflexive Banach spaces. We start with an existence and uniqueness result. We complete it with a convergence result which describes the dependence of the solution with respect to the data. To this end, we use the notion of Mosco convergence for the set of constraints. Next, we formulate an optimal control problem for which we prove the existence of optimal pairs and state a convergence result. Finally, we exemplify the use of our results in the study of a two-dimensional boundary value problem which describes the frictionless contact of an elastic body with two reactive foundations. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of nonlinear multi-valued mappings which is called a nonspreading-type mapping in Hilbert spaces, and prove some properties and the existence results for the proposed mapping. Furthermore, we prove weak and strong convergence theorems for a finite family of nonspreading-type multi-valued mappings in Hilbert spaces. As applications, we give examples and numerical results to illustrate our iteration and results. [ABSTRACT FROM AUTHOR]

In the paper, we study the analytic properties of the pair consisting of the L-functions of new forms and periodic Hurwitz zeta-functions with a transcendental parameter, i.e., we investigate the mixed joint discrete value distribution of this tuple in two cases. For this purpose, certain arithmetical conditions are used. Also, we give some applications of these limit theorems. [ABSTRACT FROM AUTHOR]

This paper is devoted to the investigation of conditions for the weak convergence in the space C(T) of the stochastic processes from the space Fψ(Ω). Using these conditions the limit theorem for stochastic processes from the space Fψ(Ω) is obtained. This theorem can be utilized for achieving the given approximation accuracy and reliability of integrals depending on parameter by Monte Carlo method. [ABSTRACT FROM AUTHOR]

We provide conditions on dependent and on non-stationary random variables X n ensuring that the mantissa of the sequence of products ∏ 1 n X k is almost surely distributed following Benford’s law or converges in distribution to Benford’s law. This is achieved through proving new generalizations of Lévy’s and Robbins’s results on distribution modulo 1 of sums of independent random variables. [ABSTRACT FROM AUTHOR]

We consider the limit distribution of a sum of strongly additive arithmetic functions with arguments runningthrough shifted primes. We obtain sufficient and necessary conditions for the weak convergence of distributions of such sums to the discrete uniform law. We study the case where the functions take values 0 or 1 on primes. [ABSTRACT FROM AUTHOR]

We study the convergence of iterates of quadratic stochastic operators that are mean monotonic. They are defined on the convex set of probability measures concentrated on a weakly compact order interval S=[0,f] of a fixed Banach lattice F. We study their regularity and identify the limits of trajectories either as the “infimum” or “supremum” of the support of initial distributions. [ABSTRACT FROM AUTHOR]

We offer an umbrella type result which extends weak convergence of classical empirical processes on the line to that of more general processes indexed by functions of bounded variation. This extension is not contingent on the type of dependence of the underlying sequence of random variables. As a consequence we establish weak convergence for stationary empirical processes indexed by general classes of functions under α -mixing conditions. [ABSTRACT FROM AUTHOR]

In infinite-dimensional Hilbert spaces, we prove that the iterative sequence generated by the extragradient method for solving pseudo-monotone variational inequalities converges weakly to a solution. A class of pseudo-monotone variational inequalities is considered to illustrate the convergent behavior. The result obtained in this note extends some recent results in the literature; especially, it gives a positive answer to a question raised in Khanh (Acta Math Vietnam 41:251–263, 2016). [ABSTRACT FROM AUTHOR]

In this paper, we suggest and analyze a Krasnoselski-Mann type iterative method to approximate a common element of solution sets of a hierarchical fixed point problem for nonexpansive mappings and a split mixed equilibrium problem. We prove that sequences generated by the proposed iterative method converge weakly to a common element of solution sets of these problems. Further, we derive some consequences from our main result. Furthermore, we extend the considered iterative method to a split monotone variational inclusion problem and deduce some consequences. Finally, we give a numerical example to justify the main result. The method and results presented in this paper generalize and unify the corresponding known results in this area. [ABSTRACT FROM AUTHOR]

Rosenthal type moment inequalities of negative superadditive dependent (NSD) random arrays and truncation method are used to research weak convergence, Lr convergence and complete convergence of line weighting and maximum of negative superadditive dependent random array on the basis of the unanimous integrable condition of real number array. The results obtained can generalize and improve the convergence of the existing negative dependent array (NA), the maximum value and the truncated (END) sequence. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish sharp estimates for the moduli of continuity of the local time of a class of sub-fractional Brownian motions. We also investigate the continuity of their local times with respect to the self-similarity index. [ABSTRACT FROM AUTHOR]

This paper shows how the theory of Dirichlet forms can be used to deliver proofs of optimal scaling results for Markov chain Monte Carlo algorithms (specifically, Metropolis–Hastings random walk samplers) under regularity conditions which are substantially weaker than those required by the original approach (based on the use of infinitesimal generators). The Dirichlet form methods have the added advantage of providing an explicit construction of the underlying infinite-dimensional context. In particular, this enables us directly to establish weak convergence to the relevant infinite-dimensional distributions. [ABSTRACT FROM AUTHOR]

We know that variational inequality problem is very important in the nonlinear analysis. For a variational inequality problem defined over a nonempty fixed point set of a nonexpansive mapping in Hilbert space, the strong convergence theorem has been proposed by I. Yamada. The algorithm in this theorem is named the hybrid steepest descent method. Based on this method, we propose a new weak convergence theorem for zero points of inverse strongly monotone mapping and fixed points of nonexpansive mapping in Hilbert space. Using this result, we obtain some new weak convergence theorems which are useful in nonlinear analysis and optimization problem. [ABSTRACT FROM AUTHOR]

In this paper, we prove that an isotropic complex symmetric α-stable random measure ( $0<\alpha<2$ ) can be approximated by a complex process constructed by integrals based on the Poisson process with random intensity. [ABSTRACT FROM AUTHOR]

In this paper, an iterative algorithm investigated for m-accretive and inverse-strongly accretive operators. Also, a weak convergence theorem for the sum of two accretive operators is established in a real uniformly convex and q-uniformly smooth Banach space. [ABSTRACT FROM AUTHOR]

In this paper, we study the split equality feasibility problem and present two algorithms for solving the problem with special structure. We prove the weak convergence of these algorithms under mild conditions. Especially, the selection of stepsize is only dependent on the information of current iterative points, but independent from the prior knowledge of operator norms. These algorithms provide new ideas for solving the split equality feasibility problem. Numerical results demonstrate the feasibility and effectiveness of these algorithms. [ABSTRACT FROM AUTHOR]

We consider the limit distribution of values of a sum of sets of strongly additive arithmetic functions with shifted argument. We obtain sufficient and necessary conditions for a weak convergence of distributions of that sum to the discrete uniform law. The case where those functions take values 0 or 1 on primes is studied. [ABSTRACT FROM AUTHOR]

The Douglas-Rachford algorithm is a very popular splitting technique for finding a zero of the sum of two maximally monotone operators. The behaviour of the algorithm remains mysterious in the general inconsistent case, i.e., when the sum problem has no zeros. However, more than a decade ago, it was shown that in the (possibly inconsistent) convex feasibility setting, the shadow sequence remains bounded and its weak cluster points solve a best approximation problem. In this paper, we advance the understanding of the inconsistent case significantly by providing a complete proof of the full weak convergence in the convex feasibility setting. In fact, a more general sufficient condition for the weak convergence in the general case is presented. Our proof relies on a new convergence principle for Fejér monotone sequences. Numerous examples illustrate our results. [ABSTRACT FROM AUTHOR]

The Krasnoselskii-Mann iteration plays an important role in the approximation of fixed points of nonexpansive operators; it is known to be weakly convergent in the infinite dimensional setting. In this present paper, we provide a new inexact Krasnoselskii-Mann iteration and prove weak convergence under certain accuracy criteria on the error resulting from the inexactness. We also show strong convergence for a modified inexact Krasnoselskii-Mann iteration under suitable assumptions. The convergence results generalize existing ones from the literature. Applications are given to the Douglas-Rachford splitting method, the Fermat-Weber location problem as well as the alternating projection method by John von Neumann. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study iterative algorithms for solving split mixed equilibrium problems and fixed point problems of λ-hybrid multivalued mappings in real Hilbert spaces and prove that the proposed iterative algorithm converges weakly to a common solution of the considered problems. We also provide an example to illustrate the convergence behavior of the proposed iteration process. [ABSTRACT FROM AUTHOR]

A general approach to derive the weak convergence, when centered and rescaled, of certain Bayesian nonparametric priors is proposed. This method may be applied to a wide range of processes including, for instance, nondecreasing nonnegative pure jump Lévy processes and normalized nondecreasing nonnegative pure jump Lévy processes with known finite dimensional distributions. Examples clarifying this approach involve the beta process in latent feature models and the Dirichlet process. [ABSTRACT FROM AUTHOR]