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The convective Brinkman-Forchheimer (CBF) equations are employed to characterize the motion of incompressible fluid in a saturated porous medium. This work investigates the small noise asymptotic of two-time-scale stochastic CBF equations in two and three dimensional bounded domains. More precisely, we establish a Wentzell-Freidlin type large deviation principle for stochastic partial differential equations that have slow and fast time-scales. The slow component is the stochastic CBF equations in two or three dimensions perturbed by a small multiplicative Gaussian noise, while the fast component is a stochastic reaction-diffusion equation with damping. The results are obtained by using a variational method (based on weak convergence approach) developed by Budhiraja and Dupuis, Khasminkii's time discretization approach and stopping time arguments. In particular, the findings from this study are also applicable to two-dimensional stochastic Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

Recently, the Randomized Kaczmarz algorithm (RK) draws much attention because of its low computational complexity and less requirement on computer memory. Many existing results on analysis focus on the behavior of RK in Euclidean spaces, and typically derive exponential converge rates with the base tending to one, as the condition number of the system increases. The dependence on the condition number largely restricts the application of these estimates. There are also results using relaxation (i.e., small step-sizes tending to zero as the sample size increases) to achieve polynomial convergence rates of RK in Hilbert spaces. In this paper, we prove the weak convergence (with polynomial rates) of RK with constant step-size in Hilbert spaces. As a consequence, the strong convergence of RK in Euclidean spaces is obtained with condition number-free polynomial rates. In the setting with noisy data, we study the relaxation technique and obtain a strong convergence of RK in Hilbert spaces, with the rates arbitrarily close to the minimax optimal ones. We apply the analysis to reproducing kernel-based online gradient descent learning algorithms and improve the state-of-the-art convergence estimate in the literature. [ABSTRACT FROM AUTHOR]

This article investigates the weak approximation towards the invariant measure of semi-linear stochastic differential equations (SDEs) under non-globally Lipschitz coefficients. For this purpose, we propose a linear-theta-projected Euler (LTPE) scheme, which also admits an invariant measure, to handle the potential influence of the linear stiffness. Under certain assumptions, both the SDE and the corresponding LTPE method are shown to converge exponentially to the underlying invariant measures, respectively. Moreover, with time-independent regularity estimates for the corresponding Kolmogorov equation, the weak error between the numerical invariant measure and the original one can be guaranteed with convergence of order one. In terms of computational complexity, the proposed ergodicity preserving scheme with the nonlinearity explicitly treated has a significant advantage over the ergodicity preserving implicit Euler method in the literature. Numerical experiments are provided to verify our theoretical findings. • Approximation to the invariant measure of nonglobal Lipschitz SDEs. • A new (LTPE) scheme: ergodicity-preserving and nonlinearity explicitly treated. • LTPE scheme handles the challenges posed by linear stiff system. • Weak convergence analysis of LTPE established with convergence order one. [ABSTRACT FROM AUTHOR]

Many authors have exploited the fact that the distribution of the multivariate probability integral transformation (PIT) of a continuous random vector X ∈ R d with cumulative distribution function F X is free of the marginal distributions. While most of these methods are based on the cdf of W = F X (X) , this paper introduces the weighted characteristic function (WCf) of W. A sample version of the WCf of W based on pseudo-observations is proposed and its weak limit in a space of complex functions is formally established. This result can be used to define test statistics for multivariate independence and goodness-of-fit in copula models, whose asymptotic behaviour comes from the weak convergence of the empirical WCf process. Simulations show the good sampling properties of these new tests, and an illustration is given on the multivariate Cook and Johnson dataset. [ABSTRACT FROM AUTHOR]

Delays are ubiquitous, pervasive, and entrenched in everyday life, thus taking it into consideration is necessary. Dupire recently developed a functional Itô formula, which has changed the landscape of the study of stochastic functional differential equations and encouraged a reconsideration of many problems and applications. Based on the new development, this work examines functional diffusions with two-time scales in which the slow-varying process includes path-dependent functionals and the fast-varying process is a rapidly-changing diffusion. The gene expression of biochemical reactions occurring in living cells in the introduction of this paper is such a motivating example. This paper establishes mixed functional Itô formulas and the corresponding martingale representation. Then it develops an averaging principle using weak convergence methods. By treating the fast-varying process as a random "noise", under appropriate conditions, it is shown that the slow-varying process converges weakly to a stochastic functional differential equation whose coefficients are averages of that of the original slow-varying process with respect to the invariant measure of the fast-varying process. [ABSTRACT FROM AUTHOR]

We show how to pass to the limit in some nonlinear elliptic problems, despite the weak convergence of the right hand sides (and the weak convergence of the solutions). [ABSTRACT FROM AUTHOR]

By the Voronin theorem, the set of shifts of the Riemann zeta-function ζ (s + i τ) , s = σ + i t , τ ∈ R , that approximate any given non-vanishing analytic function defined on { s ∈ C : 1 2 < σ < 1 } has a positive lower density. In the paper, it is proved that the same property of the above shifts remains valid in the intervals [ T , T + H ] with T 1 / 3 (log ⁡ T) 26 / 15 ⩽ H ⩽ T. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with modeling the dependence structure of two (or more) time series in the presence of a (possibly multivariate) covariate which may include past values of the time series. We assume that the covariate influences only the conditional mean and the conditional variance of each of the time series but the distribution of the standardized innovations is not influenced by the covariate and is stable in time. The joint distribution of the time series is then determined by the conditional means, the conditional variances and the marginal distributions of the innovations, which we estimate nonparametrically, and the copula of the innovations, which represents the dependency structure. We consider a nonparametric and a semiparametric estimator based on the estimated residuals. We show that under suitable assumptions, these copula estimators are asymptotically equivalent to estimators that would be based on the unobserved innovations. The theoretical results are illustrated by simulations and a real data example. [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]

A common assumption in pair-copula constructions is that the copula of the conditional distribution of two random variables given a covariate does not depend on the value of that covariate. Two conflicting intuitions arise about the best possible rate of convergence attainable by nonparametric estimators of that copula. On the one hand, the best possible rates for estimating the marginal conditional distribution functions are slower than the parametric one. On the other hand, the invariance of the conditional copula given the value of the covariate suggests the possibility of parametric convergence rates. The more optimistic intuition is shown to be correct, confirming a conjecture supported by extensive Monte Carlo simulations by Hobæk Haff and Segers (2015) and improving upon the nonparametric rate obtained theoretically by Gijbels et al. (2015). The novelty of the proposed approach lies in a double smoothing procedure for the estimator of the marginal conditional distribution functions. The copula estimator itself is asymptotically equivalent to an oracle empirical copula, as if the marginal conditional distribution functions were known. [ABSTRACT FROM AUTHOR]

It is widely noted in the literature that the degradation should be simplified into a memoryless Markovian process for the purpose of predicting the remaining useful life (RUL). However, there actually exists the long-term dependency in the degradation processes of some industrial systems, including electromechanical equipments, oil tankers, and large blast furnaces. This implies the new degradation state depends not only on the current state, but also on the historical states. Such dynamic systems cannot be accurately described by traditional Markovian models. Here we present an improved non-Markovian degradation model with both the long-term dependency and the item-to-item uncertainty. As a typical non-stationary process with dependent increments, fractional Brownian motion (FBM) is utilized to simulate the fractal diffusion of practical degradations. The uncertainty among multiple items can be represented by a random variable of the drift. Based on this model, the unknown parameters are estimated through the maximum likelihood (ML) algorithm, while a closed-form solution to the RUL distribution is further derived using a weak convergence theorem. The practicability of the proposed model is fully verified by two real-world examples. The results demonstrate that the proposed method can effectively reduce the prediction error. [ABSTRACT FROM AUTHOR]

For a strictly stationary sequence of R + d –valued random vectors we derive functional convergence of partial maxima stochastic processes under joint regular variation and weak dependence conditions. The limit process is an extremal process and the convergence takes place in the space of R + d –valued càdlàg functions on [ 0 , 1 ] , with the Skorohod weak M 1 topology. We also show that this topology in general cannot be replaced by the stronger (standard) M 1 topology. The theory is illustrated on three examples, including the multivariate squared GARCH process with constant conditional correlations. [ABSTRACT FROM AUTHOR]

We construct a family of processes, from a single Poisson process, that converges in law to a complex Brownian motion. Moreover, we find realizations of these processes that converge almost surely to the complex Brownian motion, uniformly on the unit time interval. Finally the rate of convergence is derived. [ABSTRACT FROM AUTHOR]

A general result on weak convergence of the empirical measure of discretely observed functional data is shown. It is applied to the problem of estimation of functional mean value, and the problem of consistency of various types of depth for functional data. Counterexamples illustrating the fact that the assumptions as stated cannot be dropped easily are given. [ABSTRACT FROM AUTHOR]

We investigate the behaviour of population models, specified in stochastic Concurrent Constraint Programming ( sCCP ). In particular, we focus on models from which we can define a semantics both in terms of Continuous Time Markov Chains (CTMC) and in terms of Stochastic Hybrid Systems, in which some populations are approximated continuously, while others are kept discrete. We will prove the correctness of the hybrid semantics from the point of view of the limiting behaviour of a sequence of models for increasing population size. More specifically, we prove that, under suitable regularity conditions, the sequence of CTMC constructed from sCCP programs for increasing population size converges to the hybrid system constructed by means of the hybrid semantics. We investigate in particular what happens for sCCP models in which some transitions are guarded by boolean predicates or in the presence of instantaneous transitions. [ABSTRACT FROM AUTHOR]

This article studies the weak convergence and associated Central Limit Theorem for blurring and nonblurring processes. Then, they are applied to the estimation of location parameter. Simulation studies show that the location estimation based on the convergence point of blurring process is more robust and often more efficient than that of nonblurring process. [ABSTRACT FROM AUTHOR]

Let { A k } k = 1 ∞ be a sequence of finite subsets of R d satisfying that # A k ≥ 2 for all integers k ≥ 1. In this paper, we first give a sufficient and necessary condition for the existence of the infinite convolution ν = δ A 1 ⁎ δ A 2 ⁎ ⋯ ⁎ δ A n ⁎ ⋯ , where all sets A k ⊆ R + d and δ A = 1 # A ∑ a ∈ A δ a. Then we study the spectrality of a class of infinite convolutions generated by Hadamard triples in R and construct a class of singular spectral measures without compact support. Finally we show that such measures are abundant, and the dimension of their supports has the intermediate-value property. [ABSTRACT FROM AUTHOR]

This work focuses on functional differential equations subject to wideband noise perturbations. Modeling using a white noise is often an idealization of the actual physical process, whereas a wideband noise can be easily realized in applications and well approximates a white noise. Using functional derivatives together with the combined perturbed test function methods and martingale techniques, this paper demonstrates that when a small parameter tends to zero, the underlying process converges to a limit that is the solution of a stochastic functional differential equation. To illustrate, an integro-differential system with wideband noise perturbation is examined as an example. Not only are the results interesting from a mathematical point of view, but also they are of utility to a wide range of applications. [ABSTRACT FROM AUTHOR]

We show that the system of point vortices, perturbed by a certain transport type noise, converges weakly to the vorticity form of 2D Navier–Stokes equations driven by the space-time white noise. [ABSTRACT FROM AUTHOR]

We study measurable dependence of measures on a parameter in the following two classical problems: constructing conditional measures and the Kantorovich optimal transportation. For parametric families of measures and mappings we prove the existence of conditional measures measurably depending on the parameter. A particular emphasis is made on the Borel measurability (which cannot be always achieved). Our second main result gives sufficient conditions for the Borel measurability of optimal transports and transportation costs with respect to a parameter in the case where marginal measures and cost functions depend on a parameter. As a corollary we obtain the Borel measurability with respect to the parameter for disintegrations of optimal plans. Finally, we show that the Skorohod parametrization of measures by mappings can be also made measurable with respect to a parameter. [ABSTRACT FROM AUTHOR]