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This paper presents a fully adaptive iterative method without linesearch procedure to solve variational inequalities in the setting of quasi-monotone which is locally Lipschitz. Our method involves one functional evaluation alongside one projection per iteration and our stepsizes are not restricted to be nonincreasing as opposed to relevant methods without linesearch procedure but with nonincreasing stepsizes. We give weak and strong convergence analysis of the sequences generated by our proposed method under some mild conditions. We also give R -linear rate convergence under an error bound condition and give some numerical implementations of our method. Numerical results show that our proposed method is competitive with other related methods in the literature. [ABSTRACT FROM AUTHOR]

Let N t (n) denote the number of blocks in a Ξ -coalescent restricted to a sample of size n ∈ N after time t ≥ 0. Under the assumption of a certain curvature condition on a function well-known from the literature, we prove the existence of sequences (v (n , t)) n ∈ N for which (log N t (n) − log v (n , t)) t ≥ 0 converges to an Ornstein–Uhlenbeck type process as n → ∞. The curvature condition is intrinsically related to the behavior of Ξ near the origin. The method of proof is to show the uniform convergence of the associated generators. Via Siegmund duality an analogous result for the fixation line is proven. Several examples are studied. [ABSTRACT FROM AUTHOR]

In this present work, we study new algorithms to solve the split equality fixed-point problems for total quasi-asymptotically nonexpansive mappings in Hilbert spaces. In addition, we have established the convergence criteria for the proposed algorithms, at the end, we provided numerical results that justified our theoretical results. The results presented in this paper, provided a unified framework for studying this kind of problem involving different classes of mappings. [ABSTRACT FROM AUTHOR]

We consider stochastic differential equations with bounded α -Hölder continuous drift, with α ∈ (0 , 1) , driven by multiplicative noise. We moreover assume that the diffusion coefficient is uniformly elliptic and bounded, with bounded first and second order partial derivatives. We show that the weak rate of convergence for the Euler–Maruyama scheme is arbitrarily close to (1 + α) / 2. [ABSTRACT FROM AUTHOR]

This paper is devoted to numerically solving a class of optimal stopping problems for stochastic hybrid systems involving both continuous states and discrete events. The motivation for solving this class of problems stems from quickest event detection problems of stochastic hybrid systems in broad application domains. We solve the optimal stopping problems numerically by constructing feasible algorithms using Markov chain approximation techniques. The key tasks we undertake include designing and constructing discrete-time Markov chains that are locally consistent with switching diffusions, proving the convergence of suitably scaled sequences, and obtaining convergence for the cost and value functions. Finally, numerical results are provided to demonstrate the performance of the algorithms. [ABSTRACT FROM AUTHOR]

This paper focuses on a class of functional diffusion systems with singularly perturbed regime switching, where the modulating Markov chain has a large state space and undergoes weak and strong interactions. By using the martingale method and weak convergence, this paper shows that the underlying system will weakly converge to a limit system, which is simpler than the original system. For a class of integro-differential diffusion system with singularly perturbed regime switching, as a class of special functional diffusion system, this paper demonstrates that if the limit system is moment exponentially stable, the original system with singular perturbation is also moment exponentially stable under suitable conditions. This result is interesting since the limit system is always simpler. Finally, an example is given to illustrate this result. [ABSTRACT FROM AUTHOR]

We provide a unified treatment of pathwise large and moderate deviations principles for a general class of multidimensional stochastic Volterra equations with singular kernels, not necessarily of convolution form. Our methodology is based on the weak convergence approach by Budhiraja and Dupuis (2019); Dupuis and Ellis (1997). We show in particular how this framework encompasses most rough volatility models used in mathematical finance, yields pathwise moderate deviations for the first time and generalises many recent results in the literature. [ABSTRACT FROM AUTHOR]

The cumulative incidence function (CIF) plays an important role in the comparison of competing risks in a competing risks model. Its value at time t is the probability of failure by time t from a particular type of risk in the presence of other risks. In this paper we consider the estimation of two CIFs, F 1 and F 2 , corresponding to two competing risks when the ratio R (t) ≡ F 1 (t) / F 2 (t) is nondecreasing in t > 0. First, we derive their nonparametric maximum likelihood estimators (NPMLE) of these CIFs in the continuous case under this order constraint and show that they are inconsistent. We then develop projection-type estimators that are uniformly strongly consistent and study the weak convergence of the resulting processes. Through simulations, we compare the finite sample performance of the NPMLEs and our estimators and show that our estimators outperform them in general in terms of mean square error at all the scenarios that we consider. We also develop a test for the presence of this order constraint and extend all these results to the censored case. To illustrate the applicability of the theory we develop, we provide a real life example. • Uniformly consistent estimation of two cumulative incidence functions in a competing risks model when the ratio is monotonic. • Weak convergence of the estimators of two cumulative incidence functions in a competing risks model when the ratio is monotonic. • Test of equality of two cumulative incidence functions against the alternative that their ratio is monotonic in a competing risks model. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the generalized variational inequality problem in real Hilbert spaces. We propose an explicit proximal method which requires only one proximal step and one mapping evaluation per iteration and also uses an adaptive step-size rule that enables to avoid the prior knowledge of the Lipschitz constant of the involved mapping. Weak convergence of the proposed scheme is established under standard assumptions. Under strong monotonicity, we present the R -linear convergence rate of our new method. Intensive numerical experiments illustrate the advantages and the applicability of our scheme. Moreover, our work generalizes theoretically several recent results in this field. [ABSTRACT FROM AUTHOR]

We consider the modified Surface Quasi-Geostrophic (mSQG) equation on the 2D torus T 2 , perturbed by multiplicative transport noise. The equation admits the white noise measure on T 2 as the invariant measure. We first prove the existence of white noise solutions to the stochastic equation via the method of point vortex approximation, then, under a suitable scaling limit of the noise, we show that the solutions converge weakly to the unique stationary solution of the dissipative mSQG equation driven by space–time white noise. The weak uniqueness of the latter equation is also proved by following Gubinelli and Perkowski's approach in Gubinelli and Perkowski (2020). [ABSTRACT FROM AUTHOR]

In this paper, we first study a relaxed inertial projection and contraction method for variational inequalities to obtain weak convergence results under standard assumptions in Hilbert spaces. Next, we propose another inertial projection and contraction method for which the inertial factor θ is chosen in [ 0 , 1 ] with θ = 1 possible. Weak and linear convergence are obtained for this second proposed method. As far as we know, the choice θ = 1 has not been considered before in the literature for inertial projection and contraction method. Finally, we give some numerical illustrations of our methods. [ABSTRACT FROM AUTHOR]

We propose a full discretization to approximate the invariant measure numerically for parabolic stochastic partial differential equations (SPDEs) with non-globally Lipschitz coefficients. We present a priori estimates and regularity estimates of the numerical solution via a variational approach and Malliavin calculus. Under certain hypotheses, we present the time-independent regularity estimates for the corresponding Kolmogorov equation and the time-independent weak convergence analysis for the full discretization. Furthermore, we show that the V-uniformly ergodic invariant measure of the original system is approximated by this full discretization with weak convergence rate. Numerical experiments verify theoretical findings. [ABSTRACT FROM AUTHOR]

Can convergence of random variables, random vectors or, more generally, random elements of a metric space be studied using fuzzy integrals, in particular the seminormed integral? Among the positive results we present, it is shown that for large families of t-seminorms convergence in distribution and convergence in probability can be characterized via seminormed integrals. This suggests that, while fuzzy integrals were devised for working with non-additive fuzzy measures, some of their theoretical properties within the realm of probability measures can be comparable to those of the Lebesgue integral. [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 article, we consider the one-dimensional stochastic wave and heat equations driven by a linear multiplicative Gaussian noise which is white in time and behaves in space like a fractional Brownian motion with Hurst index H ∈ (1 4 , 1). We prove that the solution of each of the above equations is continuous in terms of the index H , with respect to the convergence in law in the space of continuous functions. The proof is based on a tightness criterion on the plane and Malliavin calculus techniques in order to identify the limit law. [ABSTRACT FROM AUTHOR]

We analyse convergence of a micro–macro acceleration method for the simulation of stochastic differential equations with time-scale separation. The method alternates short bursts of path simulations with the extrapolation of macroscopic state variables forward in time. After extrapolation, a new microscopic state is constructed, consistent with the extrapolated macroscopic state, that minimises the perturbation caused by the extrapolation in a relative entropy sense. We study local errors and numerical stability of the method to prove its convergence to the full microscopic dynamics when the extrapolation time step tends to zero and the number of macroscopic state variables tends to infinity. [ABSTRACT FROM AUTHOR]

In this paper we show that solutions of two-dimensional stochastic Navier–Stokes equations driven by Brownian motion can be approximated by stochastic Navier–Stokes equations forced by pure jump noise/random kicks. [ABSTRACT FROM AUTHOR]

In traditional works on numerical schemes for solving stochastic differential equations (SDEs), the globally Lipschitz assumption is often assumed to ensure different types of convergence. In practice, this is often too strong a condition. Brownian motion driven SDEs used in applications sometimes have coefficients which are only Lipschitz on compact sets, but the paths of the SDE solutions can be arbitrarily large. In this paper, we prove convergence in probability and a weak convergence result under a less restrictive assumption, that is, locally Lipschitz and with no finite time explosion. We prove if a numerical scheme converges in probability uniformly on any compact time set (UCP) with a certain rate under a global Lipschitz condition, then the UCP with the same rate holds when a globally Lipschitz condition is replaced with a locally Lipschitz plus no finite explosion condition. For the Euler scheme, weak convergence of the error process is also established. The main contribution of this paper is the proof of n weak convergence of the normalized error process and the limit process is also provided. We further study the boundedness of the second moments of the weak limit process and its running supremum under both global Lipschitz and locally Lipschitz conditions. [ABSTRACT FROM AUTHOR]

The mass transport processes always accompany the flow phenomena and have attracted many researchers. A lot of numerical methods have been developed to study them. These numerical methods can be classified into the Eulerian and the Lagrangian approaches. The Lagrangian approach has advantages in high stability and simplicity over the Eulerian approach, but suffers from heavy computational cost. In this paper, we are mainly concerned with the trade-offs between the accuracy and computational cost when applying the random walk method, which is a Lagrangian approach for examining the mass transport scenario. We introduce a linear model to assess the accuracy of the random walk method in several computational configurations. Studies on computational parameters, i.e. the size of time step and number of particles, are conducted with the focus on estimation of the longitudinal dispersion coefficient D L in steady flows. The results show that the proposed linear model can satisfactorily explain the computational accuracy, both in sample and out-of-sample. Furthermore, we find a constant dimensionless parameter, which quantifies a generic relationship between the accuracy and the number of particles regardless of the flow and diffusion conditions. This dimensionless parameter is of theoretic value and offers guidelines for choosing the correct computational parameters to achieve the required numerical accuracy. Image 1 • We propose a linear model for assessing the accuracy of the random walk method. • A dimensionless parameter is proposed to link computational accuracy and particle number. • The present computational results are found to be insensitive to the time step in Couette flow. [ABSTRACT FROM AUTHOR]

In this paper a new distributed multi-agent Actor-Critic algorithm for reinforcement learning is proposed for solving multi-agent multi-task optimization problems. The Critic algorithm is in the form of a Distributed Emphatic Temporal Difference DETD(λ) algorithm, while the Actor algorithm is proposed as a complementary consensus based policy gradient algorithm, derived from a global objective function having the role of a scalarizing function in multi-objective optimization. It is demonstrated that the Feller-Markov properties hold for the newly derived Actor algorithm. A proof of the weak convergence of the algorithm to the limit set of an attached ODE is derived under mild conditions, using a specific decomposition between the Critic and the Actor algorithms and additional two-time-scale stochastic approximation arguments. An experimental verification of the algorithm properties is given, showing that the algorithm can represent an efficient tool for practice. [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]

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]

Abstract In applications the properties of a stochastic feature often change gradually rather than abruptly, that is: after a constant phase for some time they slowly start to vary. In this paper we discuss statistical inference for the detection and the localization of gradual changes in the jump characteristic of a discretely observed Ito semimartingale. We propose a new measure of time variation for the jump behaviour of the process. The statistical uncertainty of a corresponding estimate is analysed by deriving new results on the weak convergence of a sequential empirical tail integral process and a corresponding multiplier bootstrap procedure. [ABSTRACT FROM AUTHOR]

We consider the time-fractional nonlinear stochastic fourth-order reaction diffusion equation driven by the fractional noise (FSRDE). The FSRDE is discretized by using the mixed finite element method in space and the FBT- θ method in time. Some good techniques are proposed to prove the important lemmas in the proof of the strong convergence and the weak convergence. By proving the error estimates on the strong and weak convergence, we find the strong convergence and the weak convergence have the same convergence rate, and the convergence order is related to the fractional noise derivative β but not to the time fractional derivative α and the discrete format coefficient θ. Finally, the theorems on the strong and weak convergence are verified by the numerical tests. [ABSTRACT FROM AUTHOR]

In this work, we investigate a single projection method with double inertial extrapolation steps and self-adaptive step sizes for solving pseudomonotone variational inequality problems in a real Hilbert space. Weak and linear convergence are presented under suitable conditions. Some simple numerical examples are given to illustrate the performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This article deals with the weak error in averaging principle for a stochastic wave equation on a bounded interval [ 0 , L ] , perturbed by an oscillating term arising as the solution of a stochastic reaction–diffusion equation evolving on the fast time scale. Under suitable conditions, it is proved that the rate of weak convergence of the original solution to the solution of the corresponding averaged equation is of order 1 via an asymptotic expansion approach. [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 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]

We study a class of non-stationary shot noise processes which have a general arrival process of noises with non-stationary arrival rate and a general shot shape function. Given the arrival times, the shot noises are conditionally independent and each shot noise has a general (multivariate) cumulative distribution function (c.d.f.) depending on its arrival time. We prove a functional weak law of large numbers and a functional central limit theorem for this new class of non-stationary shot noise processes in an asymptotic regime with a high intensity of shot noises, under some mild regularity conditions on the shot shape function and the conditional (multivariate) c.d.f. We discuss the applications to a simple multiplicative model (which includes a class of non-stationary compound processes and applies to insurance risk theory and physics) and the queueing and work-input processes in an associated non-stationary infinite-server queueing system. To prove the weak convergence, we show new maximal inequalities and a new criterion of existence of a stochastic process in the space D given its consistent finite dimensional distributions, which involve a finite set function with the superadditive property. [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]

Combining inertial-type methods with projection extragradient methods, we propose three inertial extragradient algorithms for solving nonmonotone and non-Lipschitzian equilibrium problems in Hilbert spaces. Under the assumption that the solution set of the associated Minty equilibrium problem is nonempty, we establish the weak and strong convergence of the proposed algorithms, respectively. The convergence is guaranteed without any monotonicity and Lipschitz-type continuity of the equilibrium bifunction. Some numerical experiments illustrate that the presented algorithms are faster than the existing projection extragradient ones. [ABSTRACT FROM AUTHOR]