Lp Convergence Research Articles

On a weak law of large numbers and Lp -convergence for general random variables

In the present paper, we consider a sequence of the general random variables, which include the sequences of martingale differences, and establish the weak law of large numbers and the convergence in Lp under some weaker conditions.

Some notes on ergodic theorem for [formula omitted]-statistics of order [formula omitted] for stationary and not necessarily ergodic sequences

In this note, we give sufficient conditions for the almost sure and the convergence in Lp of a U-statistic of order m built on a strictly stationary but not necessarily ergodic sequence.

Trigonometric generated Lp degree of approximation

In this article we continue the study of smooth Picard singular integral operators that started in [3], see there chapters 10-14. This time the foundation of our research is a trigonometric Taylor’s formula. We establish the Lp convergence of our operators to the unit operator with rates via Jackson type inequalities engaging the first Lp modulus of continuity. Of interest here is a residual appearing term. Note that our operators are not positive.

An explicit approximation for super-linear stochastic functional differential equations

Since it is difficult to implement implicit schemes on the infinite-dimensional space, we aim to develop the explicit numerical method for approximating super-linear stochastic functional differential equations (SFDEs). Precisely, borrowing the truncation idea and linear interpolation we propose an explicit truncated Euler–Maruyama (EM) scheme for SFDEs, and obtain the boundedness and convergence in Lp(p≥2). We also prove the convergence rate with 1/2 order. Different from some previous works (Mao, 2003; Zhang et al., 2018), we release the global Lipschitz restriction on the diffusion coefficient. Furthermore, we reveal that numerical solutions preserve the underlying exponential stability. Moreover, we give several examples to support our theory.

Approximation by exponential-type polynomials

In this paper, a family of exponential-type polynomials is introduced and studied. Both the uniform and the Lp convergence are established in suitable function spaces. In the Lp-case, some estimates are also achieved using an exponentially weighted version of the p-norm. Further, a Voronovskaja type formula is proved, finding the exact order of pointwise approximation in case of continuous functions having second derivative at some points. Finally, quantitative estimates for the order of approximation are established in both the continuous and the Lp-cases in terms of the modulus of continuity and K-functionals of the involved functions. In the latter estimate, a crucial role is played by the so-called Hardy-Littlewood maximal function.

Approximation error for neural network operators by an averaged modulus of smoothness

In the present paper we establish estimates for the error of approximation (in the Lp-norm) achieved by neural network (NN) operators. The above estimates have been given by means of an averaged modulus of smoothness introduced by Sendov and Popov, also known with the name of τ-modulus, in case of bounded and measurable functions on the interval [−1,1]. As a consequence of the above estimates, we can deduce an Lp convergence theorem for the above family of NN operators in case of functions which are bounded, measurable, and Riemann integrable on the above interval. In order to reach the above aims, we preliminarily establish a number of results; among them we can mention an estimate for the p-norm of the operators, and an asymptotic type theorem for the NN operators in case of functions belonging to Sobolev spaces.

Uniform and [formula omitted] convergences for nonparametric continuous time regressions with semiparametric applications

We obtain uniform and Lp convergence rates of kernel type nonparametric estimators for the instantaneous conditional mean and variance functions of continuous time regressions, where the regressor is assumed to be a general recurrent diffusion. Our asymptotics are developed under a general set-up, with a shrinking sampling interval and an increasing time span, and without the stationarity assumption. Based on our convergence results, we develop a semiparametric inferential procedure for continuous time predictive regressions. In particular, a robust semiparametric likelihood ratio test for linear predictability is proposed, with its limit distribution established. We also apply our convergence results to obtain the asymptotics of a semiparametric maximum likelihood estimator of the drift of recurrent diffusions. In our simulation study, we examine the finite sample performance of our robust test against several existing tests in the literature. An empirical illustration is presented to test the predictability of the excess returns of two major stock indices using the commonly used dividend–price ratio and earnings–price ratio as the predictor.

Wave Operator on Compact Manifolds

On an n-dimensional (n ≥ 2) compact connected manifold without boundary, we obtain the sharp range of p to ensure the Lp convergence of the wave operator. We are not able to show the optimum result at the end point $$\left| {{1 \over p} - {1 \over 2}} \right| = {1 \over {n - 1}}$$ if n ≠ 3, but an interesting result for n = 3 is given. The main result is an extension of a result on n-dimensional Euclidean space ℝn.

Pathwise convergence under Knightian uncertainty

In this note, we will obtain the first quasi-surely convergence rate of approximation of stochastic differential equations driven by G-Brownian motion. We carry it out by making a relation between Lp and quasi-surely convergences by considering this point that in general Lp convergence does not imply quasi-surely one and neither does quasi-surely convergence. This result will show that the rate of quasi-surely convergence can not exceed that of p-th mean.

Stationary Wong–Zakai Approximation of Fractional Brownian Motion and Stochastic Differential Equations with Noise Perturbations

In this article, we introduce a Wong–Zakai type stationary approximation to the fractional Brownian motions and provide a sharp rate of convergence in Lp(Ω). Our stationary approximation is suitable for all values of H∈(0,1). As an application, we consider stochastic differential equations driven by a fractional Brownian motion with H>1/2. We provide sharp rate of convergence in a certain fractional-type Sobolev space of the approximation, which in turn provides rate of convergence for the solution of the approximated equation. This generalises some existing results in the literature concerning approximation of the noise and the convergence of corresponding solutions.

Almost everywhere convergence of Bochner-Riesz means for the Hermite operators

Let H=−Δ+|x|2 be the Hermite operator in Rn. In this paper we study almost everywhere convergence of the Bochner-Riesz means associated with H which is defined by SRλ(H)f(x)=∑k=0∞(1−2k+nR2)+λPkf(x). Here Pkf is the k-th Hermite spectral projection operator. For 2≤p<∞, we prove thatlimR→∞⁡SRλ(H)f=fa.e. for all f∈Lp(Rn) provided that λ>λ(p)/2 and λ(p)=max⁡{n(1/2−1/p)−1/2,0}. Conversely, we also show the convergence generally fails if λ<λ(p)/2 in the sense that there is an f∈Lp(Rn) for 2n/(n−1)≤p such that the convergence fails. This is in surprising contrast with a.e. convergence of the classical Bochner-Riesz means for the Laplacian. For n≥2 and p≥2 our result tells that the critical summability index for a.e. convergence for SRλ(H) is as small as only the half of the critical index for a.e. convergence of the classical Bochner-Riesz means. When n=1, we show a.e. convergence holds for f∈Lp(R) with p≥2 whenever λ>0. Compared with the classical result due to Askey and Wainger who showed the optimal Lp convergence for SRλ(H) on R we only need smaller summability index for a.e. convergence.

Connections between the Approximation Orders of Positive Linear Operators and Their Max-Product Counterparts

In this study we establish some direct connections between arbitrary positive linear operators and their corresponding nonlinear (more exactly sublinear) max-product versions, with respect to uniform and Lp convergence. There are numerous concrete examples of approximation operators, such as Bernstein-type operators, neural network operators, sampling operators and others, where the linear and the max-product versions converge both uniformly. Here, from the quantitative uniform approximation result for an arbitrary sequence of positive linear operators, we deduce by a simple general method a quantitative uniform approximation result for its max-product counterpart. We also establish convergence with respect to the Lp -norm involving the well-known K-functionals, when the supremum of the kernel is bounded from below. Our results cover the cases of bounded and unbounded domains and the case of the Kantorovich variants of the considered operators. Applications to some max-product operators are presented.

On the approximations of solutions to stochastic differential equations under polynomial condition

The subject of this paper is an analytic approximate method for a class of stochastic differential equations with coefficients that do not necessarily satisfy the Lipschitz and linear growth conditions but behave like a polynomials. More precisely, equations from the observed class have unique solutions with bounded moments and their coefficients satisfy polynomial condition. Approximate equations are defined on partitions of a time interval, and their coefficients are Taylor approximations of the coefficients of the initial equation. The rate of Lp convergence increases when degrees in Taylor approximations of coefficients increase. At the end of the paper, an example is provided to support the main theoretical result.

Some limit theorems for dependent Bernoulli random variables

We consider a sequence of correlated Bernoulli variables whose probability of success of the current trial depends conditionally on the previous trials as a linear function of the sample mean. We extend the results of Zhang and Zhang (2015) by establishing an almost sure invariance principle and a weak invariance principle in a larger setting. Moreover, we also state a Gaussian fluctuation related to an almost sure and Lp convergence for the model.

Pointwise Estimates of Solutions for the Viscous Cahn-Hilliard Equation with Inertial Term

In this paper, we study the pointwise estimates of solutions to the viscous Cahn-Hilliard equation with the inertial term in multidimensions. We use Green’s function method. Our approach is based on a detailed analysis on the Green’s function of the linear system. And we get the solution’s Lp convergence rate.

On the upper semicontinuity of a quasiconcave functional

In the recent paper [21], the second author proved a divergence-quasiconcavity inequality for the following functional D(A)=∫Tndet⁡(A(x))1n−1dx defined on the space of positive definite matrices in Lp(Tn,Sym+(n)) with zero divergence. We consider the space Xp of tensor-fields in Lp(Tn,Sym+(n)) whose divergence is a Radon measure. We endow Xp with the weak topology given by the weak convergence in Lp and the weak-⁎ convergence of the measures representing the divergence of the tensor-fields. Our main result proves the weak upper semicontinuity of the functional D(⋅) on Xp if and only if p>nn−1. We also consider the case p≤nn−1 and show that D(⋅) is upper semicontinuous along sequences satisfying additional conditions. We use the positive result to show some properties of multi-dimensional Burgers equation.

Relating notions of convergence in geometric analysis

We relate Lp convergence of metric tensors or volume convergence to a given smooth metric to intrinsic flat and Gromov–Hausdorff convergence for sequences of Riemannian manifolds. We present many examples of sequences of conformal metrics which demonstrate that these notions of convergence do not agree in general even when the sequence is conformal, gj=fj2g0, to a fixed manifold. We then prove a theorem demonstrating that when sequences of metric tensors on a fixed manifold M are bounded, (1−1∕j)g0≤gj≤Kg0, and either the volumes converge, Volj(M)→Vol0(M), or the metric tensors converge in the Lp sense, then the Riemannian manifolds (M,gj) converge in the measured Gromov–Hausdorff and volume preserving intrinsic flat sense to (M,g0).

Global Existence and Pointwise Estimates of Solutions to Generalized Kuramoto-Sivashinsky System in Multi-Dimensions

The Cauchy problem of the generalized Kuramoto-Sivashinsky equation in multi-dimensions (n ≥ 3) is considered. Based on Green’s function method, some ingenious energy estimates are given. Then the global existence and pointwise convergence rates of the classical solutions are established. Furthermore, the Lp convergence rate of the solution is obtained.

Convergence Rates to Nonlinear Diffusive Waves for Solutions to Nonlinear Hyperbolic System

This article is involved with the asymptotic behavior of solutions for nonlinear hyperbolic system with external friction. The global existence of classical solutions is proven, and Lp convergence rates are obtained. Compared with the results obtained by Hsiao and Liu, better convergence rates are obtained in this article.

On the convergence of a non-linear ensemble Kalman smoother

Ensemble methods, such as the ensemble Kalman filter (EnKF), the local ensemble transform Kalman filter (LETKF), and the ensemble Kalman smoother (EnKS) are widely used in sequential data assimilation, where state vectors are of huge dimension. Little is known, however, about the asymptotic behavior of ensemble methods. In this paper, we prove convergence in Lp of ensemble Kalman smoother to the Kalman smoother in the large-ensemble limit, as well as the convergence of EnKS-4DVAR, which is a Levenberg–Marquardt-like algorithm with EnKS as the linear solver, to the classical Levenberg–Marquardt algorithm in which the linearized problem is solved exactly.