Exact Worst-Case Convergence Rates of the Proximal Gradient Method for Composite Convex Minimization

We study the worst-case convergence rates of the proximal gradient method for minimizing the sum of a smooth strongly convex function and a non-smooth convex function, whose proximal operator is available. We establish the exact worst-case convergence rates of the proximal gradient method in this setting for any step size and for different standard performance measures: objective function accuracy, distance to optimality and residual gradient norm. The proof methodology relies on recent developments in performance estimation of first-order methods, based on semidefinite programming. In the case of the proximal gradient method, this methodology allows obtaining exact and non-asymptotic worst-case guarantees that are conceptually very simple, although apparently new. On the way, we discuss how strong convexity can be replaced by weaker assumptions, while preserving the corresponding convergence rates. We also establish that the same fixed step size policy is optimal for all three performance measures. Finally, we extend recent results on the worst-case behavior of gradient descent with exact line search to the proximal case.     

Rates of Convergence and Law of the Iterated Logarithm for U-Processes

We study in this paper some limit theorems for U-processes. We calculate rates of convergence in the central limit theorem of nondegenerate U-processes under metric entropy with bracketing condition. In application, we improve upon the law of the iterated logarithm of Arcones. All calculations use the Ossiander chaining procedure.     

多指标随机变量Marcinkiewicz大数定律完全收敛性和收敛速度

本文考虑指标在，ｄ≥１中的独立同分布随机变量序列，得到了有关大数定律的完全收敛性和收敛速度等一些结果．     

关于ρ-混合序列对数律的收敛速度

本文研究了ρ-混合序列对数律的收敛速度，在较弱的矩条件下得到了与独立同分布实随机变量类似的结果，并获得了ρ－混合序列满意对数律的一个充分性结果；讨论了ρ－混合序列重对数律的收敛速度的问题，得到了一个重对数律的充分性条件。     

Convergence to Stable Laws in Relative Entropy

Convergence to stable laws in relative entropy is established for sums of i.i.d. random variables.     

Convergence Rates to Equilibrium for Neutron Chain Fissions

We study a class of nonlinear equations arising in the stochastic theory of neutron transport. After proving existence and uniqueness of the solution, we consider the large-time behaviour of the solution and give explicit rates of convergence of the solution towards the asymptotic state.     

Rates of Convergence to Self-Similar Solutions of Burgers' Equation

We study the large-time behavior of solutions to Burgers' equation with localized initial conditions. Previous studies have demonstrated that these solutions converge to a self-similar asymptotic solution  Θ( x, t )  with an error whose   L_p   norm is of order   t ^{−1+1/2 p}   . Noting that the temporal and spatial translational invariance of the underlying equations leads to a two-parameter family of self-similar solutions  Θ( x − x _*, t + t _*)  , we demonstrate that the optimal choice of   x _*  and   t _*  reduces the   L_p   error to the order of   t ^{−2+1/2 p}   .     

Precise Rates in the Law of the Iterated Logarithm for the First Moment Convergence

In this paper, for the partial sums with their maximums of a sequence of i.i.d. random variables, we show the precise rates in the general law of the iterated logarithm of a kind of weighted infinite series of the first moment convergence by using the strong approximation and point out the equivalent moment conditions.     

Convergence Rates for Logspline Tomography

We consider bivariate logspline density estimation for tomography data. In the usual logspline density estimation for bivariate data, the logarithm of the unknown density function is estimated by tensor product splines, the unknown parameters of which are given by maximum likelihood. In this paper we use tensor product B-splines and the projection-slice theorem to construct the logspline density estimators for tomography data. Rates of convergence are established for log-density functions assumed to belong to a Besov space.     

Rates of Convergence for Bivariate Extremes

Under a second order regular variation condition, rates of convergence of the distribution of bivariate extreme order statistics to its limit distribution are given both in the total variation metric and in the uniform metric.     

Rates of Convergence for Double Sequences

In 1900 Pringsheim presented the following definition for convergence of a double sequence (i.e. ordinary infinite matrices). A double sequence [x] has limit L if the terms of the double sequence approaches L as both the column and row indices increases. Using this notion for convergence I will present definitions for asymptotically equivalent double sequences, rate preserving four dimensional matrix transformation, and these definitions shall be used to present two natural invariance theorems.AMS Subject Classification (2000): Primary 42B15, Secondary 40C05     

Convergence Rates of Attractive-Repulsive MCMC Algorithms

Methodology and Computing in Applied Probability - We consider MCMC algorithms for certain particle systems which include both attractive and repulsive forces, making their convergence analysis...     

Rates of Convergence for Random Walk Summation

We prove a Marcinkiewicz-Zygmund type strong law of large numbersfor random walk summation methods. We show that the rate ofconvergence of this type of sums is equivalent to the existenceof moments of the summands.     

Refinement of Convergence Rates for Tail Probabilities

Let X₁, X₂,... be, i.i.d. random variables, and put . We find necessary and sufficient moment conditions for , where δ≥ 0 and q>0, and with a_n>0 and b_n is either or The series f(x) we deal with are classical series studied by Hsu and Robbins, Erdős, Spitzer, Baum and Katz, Davis, Lai, Gut, etc     

Rates of Ideal Convergence for Approximation Operators

In this paper, we study a general Korovkin-type approximation theory by using the notion of ideal convergence which includes many convergence methods, such as, the usual convergence, statistical convergence, A-statistical convergence, etc. We mainly compute the rate of ideal convergence of sequences of positive linear operators.     

Rates of Convergence for a Class of Rank Tests for Independence

An almost optimal rate of convergence estimate is obtained for a large class of rank statistics for testing independence, including Gini's and Spearman's rank correlation coefficients as well as Spearman's footrule. Bibliography: 18 titles.     

Convergence in Law to Operator Fractional Brownian Motions

In this paper, we provide two approximations in law of operator fractional Brownian motions. One is constructed by Poisson processes, and the other generalizes a result of Taqqu (Z. Wahrscheinlichkeitstheor. Verw. Geb. 31:287–302, 1975).     

Geometric Convergence Rates for Time-Sampled Markov Chains

We consider time-sampled Markov chain kernels, of the form P = _n n P ⁿ . We prove bounds on the total variation distance to stationarity of such chains. We are motivated by the analysis of near-periodic MCMC algorithms.