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We propose a finite element discretization for the steady, generalized Navier-Stokes equations for fluids with shear-dependent viscosity, completed with inhomogeneous Dirichlet boundary conditions and an inhomogeneous divergence constraint. We establish (weak) convergence of discrete solutions as well as a priori error estimates for the velocity vector field and the scalar kinematic pressure. Numerical experiments complement the theoretical findings. [ABSTRACT FROM AUTHOR]

We establish a large deviation principle (LDP) for a class of stochastic porous media equations driven by Lévy-type noise on a σ-finite measure space (E,B(E),μ), with the Laplacian replaced by a negative definite self-adjoint operator. One of the main contributions of this paper is that we do not assume the compactness of embeddings in the corresponding Gelfand triple, and to compensate for this generalization, a new procedure is provided. This is the first paper to deal with LDPs for stochastic evolution equations with Lévy noise without compactness conditions. The coefficient Ψ is assumed to satisfy nondecreasing Lipschitz nonlinearity, so an important physical problem covered by this case is the Stefan problem. Numerous examples of negative definite self-adjoint operators are applicable to our results, for example, for open E⊂Rd, L= Laplacian or fractional Laplacians, i.e., L=-(-Δ)α, α∈(0,1], generalized Schrödinger operators, i.e., L=Δ+2∇ρ/ρ⋅∇, Laplacians on fractals is also included. [ABSTRACT FROM AUTHOR]

This paper derives a free analogue of the Euler-Maruyama method (fEMM) to numerically approximate solutions of free stochastic differential equations (fSDEs). Simply speaking, fSDEs are SDEs in the context of noncommutative random variables (e.g., large random matrices). By applying the theory of multiple operator integrals, we derive a free Itô formula from Taylor expansion of operator-valued functions. Iterating the free Itô formula allows us to motivate and define fEMM. Then we consider weak and strong convergence in the fSDE setting and prove strong convergence order of 1/2 and weak convergence order of 1. Numerical examples support the theoretical results and show solutions for equations where no analytical solution is known. [ABSTRACT FROM AUTHOR]

Each sequence f1, f2, ... of random variables satisfying limM→∞ (M supk∈N P(|fk| > M)) = 0 contains a subsequence fk1, fk2, ??? which, along with all its subsequences, satisfies the weak law of large numbers limN→∞ ((1/N) ∑n=1N fkn -- DN) = 0 in probability. Here, DN is a "corrector" random variable with values in [--N, N] for each N ∈ N; these correctors are all equal to zero if, in addition, lim infn→∞ E (fn² 1{|fn|≤M})) = 0 for every M ∈ (0, ∞). [ABSTRACT FROM AUTHOR]

In the present paper, we propose a local discontinuous Galerkin approximation for fully nonhomogeneous systems of p-Navier--Stokes type. On the basis of the primal formulation, we prove well-posedness, stability (a priori estimates), and weak convergence of the method. To this end, we propose a new discontinuous Galerkin discretization of the convective term and develop an abstract nonconforming theory of pseudomonotonicity, which is applied to our problem. We also use our approach to treat the p-Stokes problem. [ABSTRACT FROM AUTHOR]

We consider stochastic inviscid dyadic models with energy-preserving noise. It is shown that the models admit weak solutions which are unique in law. Under a certain scaling limit of the noise, the stochastic models converge weakly to a deterministic viscous dyadic model, for which we provide explicit convergence rates in terms of the parameters of noise. A central limit theorem underlying such scaling limit is also established. In case that the stochastic dyadic model is viscous, we show the phenomenon of dissipation enhancement for suitably chosen noise. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the optimal control problems for a class of systems with fast-slow processes. The problem under consideration is to minimize a functional subject to a system described by a two-time scaled McKean-Vlasov stochastic differential equation whose coefficients depend on state components and their probability distributions. Firstly, we establish the existence and uniqueness of the invariant probability measure for the fast process. Then, by using the relaxed control representation and the martingale method, we prove the weak convergence of the slow process and control process in the original problem, and we obtain an associated limit problem in which the coefficients are determined by the average of those of the original problem with respect to the invariant probability measure. Finally, by establishing the nearly optimal control of the limit problem, we obtain the near optimality of the original problem. [ABSTRACT FROM AUTHOR]

Stochastic models of chemical reaction networks are an important tool to describe and analyze noise effects in cell biology. When chemical species and reaction rates in a reaction system have different orders of magnitude, the associated stochastic system is often modeled in a multiscale regime. It is known that multiscale models can be approximated with a reduced system such as mean field dynamics or hybrid systems, but the accuracy of the approximation remains unknown. In this paper, we estimate the probability distribution of low copy species in multiscale stochastic reaction systems under a short timescale. We also establish an error bound for this approximation. Throughout the paper, typical mass-action systems are mainly handled, but we also show that the main theorem can be extended to general kinetics, which generalizes existing results in the literature. Our approach is based on a direct analysis of the Kolmogorov equation, in contrast to classical approaches in the existing literature. [ABSTRACT FROM AUTHOR]

We study the rate of weak convergence of Markov chains to diffusion processes under quite general assumptions. We give an example in the financial framework, applying the convergence analysis to a multiple jumps tree approximation of the CIR process. Then, we combine the Markov chain approach with other numerical techniques in order to handle the different components in jump-diffusion coupled models. We study the analytical speed of convergence of this hybrid approach and provide an example in finance, applying our results to a tree-finite difference approximation in the Heston and Bates models. [ABSTRACT FROM AUTHOR]

The classical Fatou lemma states that the lower limit of a sequence of integrals of functions is greater than or equal to the integral of the lower limit. It is known that Fatou's lemma for a sequence of weakly converging measures states a weaker inequality because the integral of the lower limit is replaced with the integral of the lower limit in two parameters, where the second parameter is the argument of the functions. In the present paper, we provide sufficient conditions when Fatou's lemma holds in its classical form for a sequence of weakly converging measures. The functions can take both positive and negative values. Similar results for sequences of setwise converging measures are also proved. We also put forward analogies of Lebesgue's and the monotone convergence theorems for sequences of weakly and setwise converging measures. The results obtained are used to prove broad sufficient conditions for the validity of optimality equations for average-cost Markov decision processes. [ABSTRACT FROM AUTHOR]

We treat a fairly broad class of financial models that includes markets with proportional transaction costs. We consider an investor with cumulative prospect theory preferences and a nonnegativity constraint on portfolio wealth. The existence of an optimal strategy is shown in this context for a class of generalized strategies. [ABSTRACT FROM AUTHOR]

Machine learning, and in particular neural network models, have revolutionized fields such as image, text, and speech recognition. Today, many important real-world applications in these areas are driven by neural networks. There are also growing applications in engineering, robotics, medicine, and finance. Despite their immense success in practice, there is limited mathematical understanding of neural networks. This paper illustrates how neural networks can be studied via stochastic analysis and develops approaches for addressing some of the technical challenges which arise. We analyze one-layer neural networks in the asymptotic regime of simultaneously (a) large network sizes and (b) large numbers of stochastic gradient descent training iterations. We rigorously prove that the empirical distribution of the neural network parameters converges to the solution of a nonlinear partial differential equation. This result can be considered a law of large numbers for neural networks. In addition, a consequence of our analysis is that the trained parameters of the neural network asymptotically become independent, a property which is commonly called "propagation of chaos"". [ABSTRACT FROM AUTHOR]

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]

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 consider the stochastic Langevin equation with additive noises, which possesses both conformal symplectic geometric structure and ergodicity. We propose a methodology of constructing high weak order conformal symplectic schemes by converting the equation into an equivalent autonomous stochastic Hamiltonian system and modifying the associated generating function. To illustrate this approach, we construct a specific second order numerical scheme and prove that its symplectic form dissipates exponentially. Moreover, for the linear case, the proposed scheme is also shown to inherit the ergodicity of the original system, and the temporal average of the numerical solution is a proper approximation of the ergodic limit over long time. Numerical experiments are given to verify these theoretical results. [ABSTRACT FROM AUTHOR]

Stochastic ambiguity provides a rich class of uncertainty models that includes those in stochastic, robust, risk-based, and semi-infinite optimization and that accounts for uncertainty about parameter values as well as incompleteness of the description of uncertainty. We provide a novel, unifying perspective on optimization under stochastic ambiguity that rests on two pillars. First, ambiguity is formulated in terms of the (cumulative) probability distribution associated with the random elements; more specifically, ambiguity is expressed by letting this distribution belong to a subfamily of distributions that might, or might not, depend on the decision variable. We derive a series of estimates by introducing a metric for the space of distribution functions based on the hypo-distance between upper semicontinuous functions. In the process, we show that this metric is consistent with convergence in distribution (= weak* convergence) of the associated probability measures. Second, we rely on the theory of lopsided convergence to establish existence, convergence, and approximation of solutions of optimization problems with stochastic ambiguity. For the first time, we estimate a distance between bifunctions and show that this leads to bounds on the solution quality for problems with stochastic ambiguity. Among other consequences, these results facilitate the study of the "price of robustness" and related quantities. [ABSTRACT FROM AUTHOR]

Given a Banach space (E, ∥ · ∥ ) having the bounded approximation property, we provide a paradigm that can be used to establish large deviation principles for sequences of random elements (Xn)n≧1 taking values in E. A similar paradigm is given to establish weak convergence of (Xn)n≧1 to a tight Borel measure in (E,∥ · ∥). We then make use of these tools to establish a functional limit law of the iterated logarithm for the smoothed empirical process in Hölder topology. [ABSTRACT FROM AUTHOR]

We compare Wiener chaos and stochastic collocation methods for linear advectionreaction- diffusion equations with multiplicative white noise. Both methods are constructed based on a recursive multistage algorithm for long-time integration. We derive error estimates for both methods and compare their numerical performance. Numerical results confirm that the recursive multistage stochastic collocation method is of order Δ (time step size) in the second-order moments while the recursive multistage Wiener chaos method is of order ΔN +Δ² (N is the order of Wiener chaos) for advection-diffusion-reaction equations with commutative noises, in agreement with the theoretical error estimates. However, for noncommutative noises, both methods are of order one in the second-order moments. [ABSTRACT FROM AUTHOR]

A new characterization of sufficient conditions for the Lie-Trotter splitting to capture the numerical invariant measure of nonlinear ergodic Langevin dynamics up to an arbitrary order is discussed. Our characterization relies on backward error analysis and needs weaker assumptions than assumed so far in the literature. In particular, neither high weak order of the splitting scheme nor symplecticity are necessary to achieve high order approximation of the invariant measure of the Langevin dynamics. Numerical experiments confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

Given a Banach space (E, ∥ ∙ ∥) having the bounded approximation property, we provide a paradigm that can be used to establish large deviation principles for sequences of random elements (Xn)n≧1 taking values in E. A similar paradigm is given to establish weak convergence of (Xn)n≧1 to a tight Borel measure in (E,∥ ∙ ∥). We then make use of these tools to establish a functional limit law of the iterated logarithm for the smoothed empirical process in Hölder topology. [ABSTRACT FROM AUTHOR]

We introduce new sufficient conditions for a numerical method to approximate with high order of accuracy the invariant measure of an ergodic system of stochastic differential equations, independently of the weak order of accuracy of the method. We then present a systematic procedure based on the framework of modified differential equations for the construction of stochastic integrators that capture the invariant measure of a wide class of ergodic SDEs (Brownian and Langevin dynamics) with an accuracy independent of the weak order of the underlying method. Numerical experiments confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

In many applications, one needs to evaluate a path-dependent objective functional V associated with a continuous-time stochastic process X. Due to the nonlinearity and possible lack of Markovian property, more often than not, V cannot be evaluated analytically, and only Monte Carlo simulation or numerical approximation is possible. In addition, such calculations often require the handling of stopping times, the usual dynamic programming approach may fall apart, and the continuity of the functional becomes the main issue. Denoting by h the stepsize of the approximation sequence, this work develops a numerical scheme so that an approximating sequence of path-dependent functionals Vh converges to V. By a natural division of labors, the main task is divided into two parts. Given a sequence Xh that converges weakly to X, the first part provides sufficient conditions for the convergence of the sequence of path-dependent functionals Vh to V. The second part constructs a sequence of approximations Xh using Markov chain approximation methods and demonstrates the weak convergence of Xh to X, when X is the solution of a stochastic differential equation. As a demonstration, combining the results of the two parts above, approximation of option pricing for the discrete-monitoring-barrier option underlying stochastic volatility model is provided. Different from the existing literature, the weak convergence analysis is carried out by using the Skorohod topology together with the continuous mapping theorem. The advantage of this approach is that the functional under study may be a function of stopping times, projection of the underlying diffusion on a sequence of random times, and/or maximum/minimum of the underlying diffusion. [ABSTRACT FROM AUTHOR]

This work is concerned with a singularly perturbed stochastic nonlinear wave equation with a random dynamical boundary condition. A splitting is used to establish the approximating equation of the system for a sufficiently small singular perturbation parameter. The approximating equation is a stochastic parabolic equation when the power exponent of the singular perturbation parameter is in [1/2, 1) but is a deterministic wave equation when the power exponent is in (1,+∞). Moreover, if the power exponent of a singular perturbation parameter is bigger than or equal to 1/2, the same limiting equation of the system is derived in the sense of distribution, as the perturbation parameter tends to zero. This limiting equation is a deterministic parabolic equation. [ABSTRACT FROM AUTHOR]

Inspired by recent advances in the theory of modified differential equations, we propose a new methodology for constructing numerical integrators with high weak order for the time integration of stochastic differential equations. This approach is illustrated with the constructions of new methods of weak order two, in particular, semi-implicit integrators well suited for stiff (meansquare stable) stochastic problems, and implicit integrators that exactly conserve all quadratic first integrals of a stochastic dynamical system. Numerical examples confirm the theoretical results and show the versatility of our methodology. [ABSTRACT FROM AUTHOR]

In this paper we describe a general framework for deriving modified equations for stochastic differential equations (SDEs) with respect to weak convergence. Modified equations are derived for a variety of numerical methods, such as the Euler or the Milstein method. Existence of higher order modified equations is also discussed. Ill the case of linear SDEs, using the Gaussianity of the underlying solutions, we derive an SDE which the numerical method solves exactly in the weak sense. Applications of modified equations in the numerical study of Langevin equations is also discussed. [ABSTRACT FROM AUTHOR]

We prove that the sequences generated by the Douglas-Rachford method converge weakly to zero of the sum of two maximal monotone operators using new tools introduced in recent works of Eckstein and the author. The assumption of maximal monotonicity of the sum is also removed, using a recent result of Bauschke. [ABSTRACT FROM AUTHOR]

This work analyzes distributed linear averaging within a connected network of sensors that each track the stationary distribution of an ergodic Markov chain with a slowly switching regime. Our approach is based on a two-time-scale stochastic approximation. A hyperparameter modeled as a Markov chain on a slower time-scale modulates the regime of each observed Markov chain. The average of all currently observed stationary distributions constitutes the average-consensus estimate to be reached by all sensors. Assuming the Markov chains do not share a common stationary distribution conditioned on their regime, then under the proposed linear averaging algorithm, the exchange graph conditions required for the sequence of sensor state values to converge weakly to the average-consensus are obtained. Estimation of a weighted average of all observed stationary distributions, not only the current ones, is proved feasible over a long-run time horizon, provided an additional communication condition holds. The sensor state values are also shown to converge weakly to solutions of a differential inclusion when the communication exchange graphs or observed Markov chains belong to a family of possible values, thus leading to a set-valued consensus formation. The rate of convergence of the consensus algorithm is studied by considering the scaled tracking errors when oriented about their steady-state for each regime of the hyperparameter. In addition, a Brownian bridge limit is obtained for a centered and scaled sequence of empirical measures. An adaptation rate is proposed as the minimum exponential rate of the sensor trajectories to the average-consensus estimate. Various optimization problems related to this adaptation rate are posed, as well as an approximate ratio that relates between any two sets of exchange graphs the adaptation rate, sensor scaled error, and absolute sum total averaging weights. Simulations illustrate our results and observation model. [ABSTRACT FROM AUTHOR]

In this paper we deal with the von Neumann alternating projection method xk+1 = PAPBxk and with its generalization of the form xk+1 = PA(xk + λk(PAPBxk - xk)), where A, B are closed and convex subsets of a Hilbert space H and FixPAPB ≠ Ø. We do not suppose that A ∩ B ≠ Ø. We give sufficient conditions for the weak convergence of the sequence (xk) to FixPAPB in the general case and in the case A is a closed affine subspace. We present also the results of preliminary numerical experiments. [ABSTRACT FROM AUTHOR]

A modified discrete limit theorem in the sense of the weak convergence of probability measures in the space of analytic functions for the Matsumoto zeta-function is proved. [ABSTRACT FROM AUTHOR]

After a brief survey of basic results about finite-dimensional approximation settings (such as mathematical frameworks of various projection methods, including the least-squares method, the dual least-squares method, and the Galerkin method) for infinite-dimensional Moore-Penrose inverses, this paper proceeds to a detailed study from the following aspects: For projection methods, we investigate convergence and weak convergence of their approximation setting to develop a unified theory on projection methods for infinite-dimensional Moore-Penrose inverses; this investigation yields a fundamental convergence theorem (Theorem 2.2), from which the criterion for convergence, the criterion for weak convergence, and the generalized dual least-squares method are derived. We also derive general results on the least-squares method, by which two flaws in Groestch's results are corrected. For nonprojection methods (whose approximation setting is a more general framework), we investigate weak perfect convergence of their approximation setting and provide a necessary and sufficient condition of such convergence holding (Theorem 3.2). Several examples are proposed as counterexamples to illustrate the differences between some important concepts or as concrete algorithms to show how the present work can help to analyze their behavior. [ABSTRACT FROM AUTHOR]

We present an easy to implement drift splitting numerical method for the approximation of stiff, nonlinear stochastic differential equations (SDEs). The method is an adaptation of the semi-implicit backward differential formula (SBDF) multistep method for deterministic differential equations and allows for a semi-implicit discretization of the drift term to remove high order stability constraints associated with explicit methods. For problems with small noise, of amplitude ε, we prove that the method converges strongly with order O(Δt² + ε Δt + ε²Δt1/2) and thus exhibits second order accuracy when the time step is chosen to be on the order of ε or larger. We document the performance of the scheme with numerical examples and also present as an application a discretization of the stochastic Cahn-Hilliard equation which removes the high order stability constraints for explicit methods. [ABSTRACT FROM AUTHOR]

This work develops a filtering scheme for hybrid systems. The process dictating the configuration or regimes is a continuous-time Markov chain with a finite state space. Exploiting hierarchical structure of the underlying system, the states of the Markov chain are divided into a number of groups so that it jumps rapidly within each group and slowly among different groups. Focusing on reduction of computational complexity, the filtering scheme includes the following steps: (1) partition the state space of the Markov chain into subspaces, (2) derive a limit system in which the states are averaged out with respect to the invariant distributions of the Markov chain, (3) use the limit system to design quadratic variation test statistics, and (4) use the test statistics to identify which ergodic class the aggregated process belongs to and to construct near-optimal filter. For demonstration, a numerical example is presented. [ABSTRACT FROM AUTHOR]

In this paper we present weak convergence results for Markov switched increment processes toward a compound Poisson process. The switching Markov process is considered also with an asymptotic split phase space. These results are obtained by a semimartingale approach. [ABSTRACT FROM AUTHOR]

We consider the problem of a posteriori change-point detection for a sequence of independent identically distributed random variables. We propose to use $d$-risks instead of error of the first type and error of the second type. We construct an asymptotically optimal test minimizing one $d$-risk and guaranteeing another. [ABSTRACT FROM AUTHOR]

The main objective of this paper consists in discussing the concept of weak solutions of a certain type of backward stochastic differential equations. Using weak convergence in the Meyer--Zheng topology, we shall give a general existence result. The terminal condition $H$ depends in functional form on a driving cadlag process $X$, and the coefficient $f$ depends on time $t$ and in functional form on $X$ and the solution process $Y$. The functional $f(t,x,y),(t,x,y)\in [0,T]\times D([0,T];{\bf R}^{d+m})$ is assumed to be bounded and continuous in $(x,y)$ on the Skorokhod space $D([0,T]\,;{\bf R}^{d+m})$ in the Meyer--Zheng topology. By several examples of Tsirelson type, we will show that there are, indeed, weak solutions which are not strong, i.e., are not solutions in the usual sense. We will also discuss pathwise uniqueness and uniqueness in law of the solution and conclude, similar to the Yamada--Watanabe theorem, that pathwise uniqueness and weak existence ensure the existence of a (uniquely determined) strong solution. Applying these concepts, we are able to state the existence of a (unique) strong solution if, additionally to the assumptions described above, $f$ satisfies a certain generalized Lipschitz-type condition. [ABSTRACT FROM AUTHOR]

The goal of this article is to determine how the oscillation and concentration effects developed by a sequence of functions in Rd are modified by the action of sampling and reconstruction operators on regular grids. Our analysis is performed in terms of Wigner and defect measures, which provide a quantitative description of the high-frequency behavior of bounded sequences in L2 (Rd). We actually present explicit formulas that make possible the computation of such measures for sampled/reconstructed sequences. As a consequence, we are able to characterize sampling and reconstruction operators that preserve or filter the high-frequency behavior of specific classes of sequences. The proofs of our results rely on the construction and manipulation of Wigner measures associated to sequences of discrete functions. [ABSTRACT FROM AUTHOR]

Conditions are given for the viability and the weak convergence of an inexact, relaxed proximal point algorithm for finding a common zero of countably many cohypomonotone operators in a Hilbert space. In turn, new convergence results are obtained for an extended version of the proximal method of multipliers in nonlinear programming. [ABSTRACT FROM AUTHOR]

This paper introduces a general implicit iterative method for finding zeros of a maximal monotone operator in a Hilbert space which unifies three previously studied strategies: relaxation, inertial type extrapolation and projection step. The first two strategies are intended to speed up the convergence of the standard proximal point algorithm, while the third permits one to perform inexact proximal iterations with fixed relative error tolerance. The paper establishes the global convergence of the method for the weak topology under appropriate assumptions on the algorithm parameters. [ABSTRACT FROM AUTHOR]

Weak convergence of nonrandomly centered sums of independent random variables with a random number of summands is investigated under the assumption that the number of summands and the summands themselves are independent and the summands are uniformly asymptotically negligible. The theorems proved in this paper are analogues of well-known limit theorems for sums of independent random variables with a nonrandom number of summands. [ABSTRACT FROM AUTHOR]

An asymptotic decision rule of testing for the independence of components of a random vector is suggested. The rule is based on ranking of linear coordinates of observations and on application of Roy's [ABSTRACT FROM AUTHOR]

Inverse estimation concerns the recovery of an unknown input signal from blurred observations on a known transformation of that signal. The estimators considered in this paper are based on a regularized inverse of the transformation involved, employing a Hilbert space set-up. We focus on properties related to weak convergence. It is shown that linear functionals can be efficiently estimated in the Hájek-LeCam sense, provided they remain restricted to a suitable class. Outside this class, rates different from n are possible. By way of an example we present the "convolution theorem" for a deconvolution. [ABSTRACT FROM AUTHOR]

In this paper a stochastic control model is studied that admits a diffusion approximation. In the prelimit model the disturbances are given by noise processes of various types: additive stationary noise, rapidly oscillating processes, and discontinuous processes with large intensity for jumps of small size. We show that a feedback control that satisfies a Lipschitz condition and is δ-optimal for the limit model remains δ-optimal also in the prelimit model. The method of proof uses the technique of weak convergence of stochastic processes. The result that is obtained extends a previous work by the authors, where the limit model is deterministic. [ABSTRACT FROM AUTHOR]

In this paper we shall establish some results on weak convergence for vector-valued continuous random processes, among which are vector-valued sub-Gaussian processes, vector-valued random processes satisfying Lipschitz conditions, and random Fourier series with vector-valued coefficients. The results obtained can be considered as corresponding extensions of those in the real setting. Our basic ideas stem from Fernique's recent works. [ABSTRACT FROM AUTHOR]

The shift-compactness of random sums S[SUP(n),SUBN[SUBn]], S[SUP(n),SUBk] = X[SUBn,1] +…+ X[SUBn,k], of independent random variables is investigated under the assumptions that in each sum the summands and their number N[SUBn] are independent and that the summands satisfy the condition of uniform asymptotic negligibility in the form [This equation is not coverted in ascii text.] in probability for each ε > 0. Some necessary and sufficient conditions are given for the weak compactness of random sums S[SUP(n),SUBN[SUBn]] -- A[SUBn], and the form of centering constants A[SUBn] is described. [ABSTRACT FROM AUTHOR]

Communications systems often have many types of users. Since the users share the same resource, there is a conflict in their needs. This conflict leads to the imposition of controls on admission or elsewhere. In this paper, there are two types of customers, GP (Guaranteed Performance) and BE (Best Effort). We consider an admission control of GP customer which has two roles. First, to guarantee the performance of the existing GP customers, and second, to regulate the congestion for the BE users. The optimal control problem for the actual physical system is difficult. A heavy traffic approximation is used, with optimal or nearly optimal controls. It is shown that the optimal values for the physical system converge to that for the limit system and that good controls for the limit system are also good for the physical system. This is done for both the discounted and average cost per unit time cost criteria. Additionally, asymptotically, the pathwise average (not mean) costs for the physical system are nearly minimal when good nearly optimal controls for the limit system are used. Numerical data show that the heavy traffic optimal control approach can lead to substantial reductions in waiting time for BE with only quite moderate rejections of GP, under heavy traffic. It also shows that the controls are often linear in the state variables. The approach has many advantages. It is robust, simplifies the analysis (both analytical and numerical), and allows a more convenient study of the parametric dependencies. Even if optimal control is not wanted, the approach is very convenient for a systematic exploration of the possible tradeoffs among the various cost components. This is done by numerically solving a series of problems with different weights on the costs. We can then get the best tradeoffs and the control policies which give them. [ABSTRACT FROM AUTHOR]

The nonoscillatory central difference scheme of Nessyahu and Tadmor is a Godunov-type scheme for one-dimensional hyperbolic conservation laws in which the resolution of Riemann problems at the cell interfaces is bypassed thanks to the use of the staggered Lax­Friedrichs scheme. Piecewise linear MUSCL-type (monotonic upstream-centered scheme for conservation laws) cell interpolants and slope limiters lead to an oscillation-free second-order resolution. Convergence to the entropic solution was proved in the scalar case. After extending the scheme to a two-step finite volume method for two-dimensional hyperbolic conservation laws on unstructured grids, we present here a proof of convergence to a weak solution in the case of the linear scalar hyperbolic equation ut + div([This symbol cannot be presented in ASCII format] u) = 0. Since the scheme is Riemann solver­free, it provides a truly multidimensional approach to the numerical approximation of compressible flows, with a firm mathematical basis. Numerical experiments show the feasibility and high accuracy of the method. [ABSTRACT FROM AUTHOR]

We consider the construction of Markov chain approximations for an important class of deterministic control problems. The emphasis is on the construction of schemes that can be easily implemented and which possess a number of highly desirable qualitative properties. The class of problems covered is that for which the control is affine in the dynamics and with quadratic running cost. This class covers a number of interesting areas of application, including problems that arise in large deviations, risk-sensitive and robust control, robust filtering, and certain problems in computer vision. Examples are given as well as a proof of convergence. [ABSTRACT FROM AUTHOR]

Necessary and sufficient conditions are given for monotone sequences of scaled random variables with a random index to converge weakly or converge weakly with the mixing property in Rényi's sense. The main results are related with the case when the terms of the sequences are sequential maximal sums of independent random variables. [ABSTRACT FROM AUTHOR]

This paper provides necessary and sufficient conditions for the weak convergence of the distributions of sums of independent random variables to the normal distribution. The conditions are given as a combination of a part of the assumption of the classic criterion for the normal convergence and the Lyapunov condition for truncated random variables. [ABSTRACT FROM AUTHOR]