Exact Rate Of Convergence Research Articles

L 1 bounds for some martingale central limit theorems

The aim of this paper is to extend the results in [E. Bolthausen, Exact convergence rates in some martingale central limit theorems, Ann. Probab., 10(3):672–688, 1982] and [J.C. Mourrat, On the rate of convergence in the martingale central limit theorem, Bernoulli, 19(2):633–645, 2013] to the L1-distance between distributions of normalized partial sums for martingale-difference sequences and the standard normal distribution.

Rates of convergence of extreme for asymmetric normal distribution

In this paper, we derive the exact uniform convergence rate of the asymmetric normal distribution of the maximum and minimum to its extreme value limit.

A graph-theoretic approach on optimizing informed-node selection in multi-agent tracking control

A graph optimization problem for a multi-agent leader–follower problem is considered. In a multi-agent system with n followers and one leader, each agent’s goal is to track the leader using the information obtained from its neighbors. The neighborhood relationship is defined by a directed communication graph where k agents, designated as informed agents, can become neighbors of the leader. This paper establishes that, for any given strongly connected communication graph with k informed agents, all agents will converge to the leader. In addition, an upper bound and a lower bound of the convergence rate are obtained. These bounds are shown to explicitly depend on the maximal distance from the leader to the followers. The dependence between this distance and the exact convergence rate is verified by empirical studies. Then we show that minimizing the maximal distance problem is a metric k-center problem in classical combinatorial optimization studies, which can be approximately solved. Numerical examples are given to illustrate the properties of the approximate solutions.

Direct and Inverse Results on Row Sequences of Hermite–Padé Approximants

We give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of simultaneous rational interpolants with a bounded number of poles. The conditions are expressed in terms of intrinsic properties of the system of functions used to build the approximants. Exact rates of convergence for these denominators and the simultaneous rational approximants are provided.

Estimation of density level sets with a given probability content

Given a random vector X valued in ℝ d with density f and an arbitrary probability number p∈(0; 1), we consider the estimation of the upper level set<texlscub>f≥t (p)</texlscub>of f corresponding to probability content p, that is, such that the probability that X belongs to<texlscub>f≥t (p)</texlscub>is equal to p. Based on an i.i.d. random sample X 1, …, X n drawn from f, we define the plug-in level set estimate , where is a random threshold depending on the sample and [fcirc] n is a nonparametric kernel density estimate based on the same sample. We establish the exact convergence rate of the Lebesgue measure of the symmetric difference between the estimated and actual level sets.

Optimal sampling design for approximation of stochastic Itô integrals with application to the nonlinear Lebesgue integration

We study the approximation of stochastic integrals in the Itô sense. We establish the exact convergence rate of the minimal errors that can be achieved by arbitrary algorithms based on a finite number of observations of the Brownian motion. We provide a construction of optimal schemes which asymptotically attain established minimal errors. Using results obtained for the Itô integration we investigate the minimal asymptotic errors for the problem of nonlinear Lebesgue integration in the average case setting.

The Convergence Rate and Asymptotic Distribution of the Bootstrap Quantile Variance Estimator for Importance Sampling

Importance sampling is a widely used variance reduction technique to compute sample quantiles such as value at risk. The variance of the weighted sample quantile estimator is usually a difficult quantity to compute. In this paper we present the exact convergence rate and asymptotic distributions of the bootstrap variance estimators for quantiles of weighted empirical distributions. Under regularity conditions, we show that the bootstrap variance estimator is asymptotically normal and has relative standard deviation of order O(n −1/4).

The Convergence Rate and Asymptotic Distribution of the Bootstrap Quantile Variance Estimator for Importance Sampling

Importance sampling is a widely used variance reduction technique to compute sample quantiles such as value at risk. The variance of the weighted sample quantile estimator is usually a difficult quantity to compute. In this paper we present the exact convergence rate and asymptotic distributions of the bootstrap variance estimators for quantiles of weighted empirical distributions. Under regularity conditions, we show that the bootstrap variance estimator is asymptotically normal and has relative standard deviation of orderO(n−1/4).

Convergence rates of limit distribution of maxima of lognormal samples

Let Mn denote the partial maximum of an independent and identically distributed lognormal random sequence. In this paper, we derive the exact uniform convergence rate of the distribution of the normalized maximum (Mn−bn)/an to its extreme value distribution, where the constants an and bn are chosen by an optimal way.

Large deviations of the limiting distribution in the Shanks–Rényi prime number race

AbstractLet q ≥ 3, 2 ≤ r ≤ φ(q) and a1, . . ., ar be distinct residue classes modulo q that are relatively prime to q. Assuming the Generalized Riemann Hypothesis (GRH) and the Linear Independence Hypothesis (LI), M. Rubinstein and P. Sarnak [11] showed that the vector-valued function Eq;a1, . . ., ar(x) = (E(x;q,a1), . . ., E(x;q,ar)), where $E(x;q,a)= ({\log x}/{\sqrt{x}})(\phi(q)\pi(x;q,a)-\pi(x))$, has a limiting distribution μq;a1, . . ., ar which is absolutely continuous on $\mathbb{R}^r$. Furthermore, they proved that for r fixed, μq;a1, . . ., ar tends to a multidimensional Gaussian as q → ∞. In the present paper, we determine the exact rate of this convergence, and investigate the asymptotic behavior of the large deviations of μq;a1, . . ., ar.

Asymptotics of the maximal radius of an Lr-optimal sequence of quantizers

Let $P$ be a probability distribution on $\mathbb{R}^d$ (equipped with an Euclidean norm $|\cdot|$). Let $ r> 0 $ and let $(\alpha_n)_{n \geq1}$ be an (asymptotically) $L^r(P)$-optimal sequence of $n$-quantizers. We investigate the asymptotic behavior of the maximal radius sequence induced by the sequence $(\alpha_n)_{n \geq1}$ defined for every $n \geq1$ by $\rho(\alpha_n) = \max{|a|, a \in\alpha_n}$. When $\card(\supp(P))$ is infinite, the maximal radius sequence goes to $\sup{|x|, x \in\operatorname{supp}(P)}$ as $n$ goes to infinity. We then give the exact rate of convergence for two classes of distributions with unbounded support: distributions with hyper-exponential tails and distributions with polynomial tails. In the one-dimensional setting, a sharp rate and constant are provided for distributions with hyper-exponential tails.

Rates of convergence of extreme for general error distribution under power normalization

By using the theory of p-max stable laws, we study the rates of convergence of extremes for general error distribution under power normalization. We derived the exact uniform convergence rate of the distribution of maximum to its extreme value limit.

Convergence and exact solutions of spline finite strip method using unitary transformation approach

The spline finite strip method (FSM) is one of the most popular numerical methods for analyzing prismatic structures. Efficacy and convergence of the method have been demonstrated in previous studies by comparing only numerical results with analytical results of some benchmark problems. To date, no exact solutions of the method or its explicit forms of error terms have been derived to show its convergence analytically. As such, in this paper, the mathematical exact solutions of spline finite strips in the plate analysis are derived using a unitary transformation approach (abbreviated as the U-transformation method herein). These exact solutions are presented for the first time in open literature. Unlike the conventional spline FSM which involves assembly of the global matrix equation and its numerical solution, the U-transformation method decouples the global matrix equation into the one involving only two unknowns, thus rendering the exact solutions of the spline finite strip to be derived explicitly. By taking Taylor’s series expansion of the exact solution, error terms and convergence rates are also derived explicitly and compared directly with other numerical methods. In this regard, the spline FSM converges at the same rate as a non-conforming finite element, yet involving a smaller number of unknowns compared to the latter. The convergence rate is also found superior to the conventional finite difference method.

Asymptotic Properties of the R/S Statistics for Linear Processes

Let {X k ; k ≥ 1} be a linear process defined by where {ϵ i ; − ∞ <i < ∞} is a doubly infinite sequence of i.i.d. random variables, and . Denote the R/S statistic by Q(n) = R(n)/S(n), where , and are the adjust range of partial sums, the sample mean, and the sample variance, respectively. In this article, the exact moment convergence rates in the law of the iterated logarithm, the law of the logarithm and the complete convergence for R/S statistics are achieved, when may be infinite.

Beyond convergence rates: exact recovery with the Tikhonov regularization with sparsity constraints

The Tikhonov regularization of linear ill-posed problems with an ℓ1 penalty is considered. We recall results for linear convergence rates and results on exact recovery of the support. Moreover, we derive conditions for exact support recovery which are especially applicable in the case of ill-posed problems, where other conditions, e.g., based on the so-called coherence or the restricted isometry property are usually not applicable. The obtained results also show that the regularized solutions do not only converge in the ℓ1-norm but also in the vector space ℓ0 (when considered as the strict inductive limit of the spaces as n tends to infinity). Additionally, the relations between different conditions for exact support recovery and linear convergence rates are investigated. With an imaging example from digital holography the applicability of the obtained results is illustrated, i.e. that one may check a priori if the experimental setup guarantees exact recovery with the Tikhonov regularization with sparsity constraints.

On the Distribution of Zeros of Faber Polynomials

We prove a local version of the potential theoretic interpretation of the Erdos-Turan result on the distribution of zeros of complex algebraic polynomials with a given uniform norm on a closed subdisk of the open unit disk. As a consequence, we derive the exact rate of convergence of the unit counting measure for zeros of Faber polynomials with respect to a Dini-smooth Jordan arc to the equilibrium measure of the arc.

Exact Moment Convergence Rates of U-Statistics

Let U n be a U-statistic based on a symmetric kernel h(x, y) and i.i.d. samples {X, X n ; n ≥ 1}. In this article, the exact moment convergence rates in the first moment of U n are obtained, which extend previous results concerning partial sums.

On the Spectral Gap of Brownian Motion with Jump Boundary

In this paper we consider the Brownian motion with jump boundary and present a new proof of a recent result of Li, Leung and Rakesh concerning the exact convergence rate in the one-dimensional case. Our methods are dierent and mainly probabilistic relying on coupling methods adapted to the special situation under investigation. Moreover we answer a question raised by Ben-Ari and Pinsky concerning the dependence of the spectral gap from the jump distribution in a multi-dimensional setting.

ASYMPTOTIC INTEGRATION OF SECOND-ORDER NONLINEAR DIFFERENCE EQUATIONS

AbstractIn this work we analyse a nonlinear, second-order difference equation on an unbounded interval. We present new conditions under which the problem admits a unique solution that is of a particular linear asymptotic form. The results concern the general behaviour of solutions to the initial-value problem, as well as solutions with a given asymptote. Our methods involve establishing suitable complete metric spaces and an application of Banach's fixed-point theorem. For the solutions found in our two main theorems—fixed initial data and fixed asymptote, respectively—we establish exact convergence rates to solutions of the differential equation related to our difference equation. It turns out that for the asymptotic case there is uniform convergence for both the solution and its derivative, while in the other case the convergence is somewhat weaker. Two different techniques are utilized, and for each one has to employ ad-hoc methods for the unbounded interval. Of particular importance is the exact form of the operators and metric spaces formulated in the earlier sections.

Convergence Rate of Extremes for the General Error Distribution

Let {X n , n ≥ 1} be an independent, identically distributed random sequence with each X n having the general error distribution. In this paper we derive the exact uniform convergence rate of the distribution of the maximum to its extreme value limit.