Convergence In Quadratic Mean Research Articles

Statement of Retraction: Convergence in quadratic mean of averaged stochastic gradient algorithms without strong convexity nor bounded gradient

Statement of Retraction: Convergence in quadratic mean of averaged stochastic gradient algorithms without strong convexity nor bounded gradient

Limit Law for Zagreb and Wiener Indices of Random Exponential Recursive Trees

The Wiener index is the sum of distances of all pairs of nodes in a graph; and the Zagreb index is defined as the sum of squares of the degrees of nodes in a rooted tree. In this note, we calculate the first two moments of the Wiener and Zagreb indices of random exponential recursive trees (random ERTs) from two systems of recurrence relations. Then, by an application of the contraction method, we characterize the limit law for a scaled Zagreb index of ERTs. Via the martingale convergence theorem, we also show the almost sure convergence and quadratic mean convergence of an appropriately scaled Wiener index that is indicative of the distance of two randomly chosen nodes.

Nonparametric kernel estimation of conditional copula density

This paper introduces a new mathematical formula for the bivariate conditional copula density and proposes kernel-type estimators for it. We demonstrate the consistency and asymptotic normality of these estimators, which also exhibit the best quadratic mean convergence rate when the optimal theoretical bandwidth is selected.

RETRACTED ARTICLE: Convergence in quadratic mean of averaged stochastic gradient algorithms without strong convexity nor bounded gradient

We, the Editors and Publisher of Statistics, have retracted the following article: Antoine Godichon-Baggioni. Convergence in quadratic mean of averaged stochastic gradient algorithms without strong convexity nor bounded gradient. Statistics. 57:3. 2023. 637–668. doi:10.1080/02331888.2023.2213371 The author raised concern regarding an error from an inequality (9) on page 647. This inequality cannot be verified if the gradient is bounded. We have been informed in our decision-making by our policy on publishing ethics and integrity and the COPE guidelines on retractions. The retracted article will remain online to maintain the scholarly record, but it will be digitally watermarked on each page as “Retracted”.

Convergence results for stochastic split feasibility problem using Perturbed Mann iterative scheme in Hilbert space

The focus of this study is on the stochastic split feasibility problem as an extension to deterministic (classical) split feasibility problem. Stochastic convergence in the quadratic mean of an iteration based on Perturbed Mann-type iterative scheme for solving the stochastic split feasibility problem is proven. The Lipschitzian and Pseudo-contractive mapping of the operators in Hilbert space was adopted. A convergence in quadratic mean to a random fixed-point of a split feasibility problem was achieved.

On the rates of convergence of parallelized averaged stochastic gradient algorithms

ABSTRACTThe growing interest for high-dimensional and functional data analysis led in the last decade to important research developing a consequent amount of techniques. Parallelized algorithms, which consist of distributing and treat the data into different machines, for example, are a good answer to deal with large samples taking values in high-dimensional spaces. We introduce here a parallelized averaged stochastic gradient algorithm, which enables to treat efficiently and recursively the data, and so, without taking care if the distribution of the data into the machines is uniform. The rate of convergence in quadratic mean, as well as the asymptotic normality of the parallelized estimates are given, for strongly and locally strongly convex objectives.

Online estimation of the asymptotic variance for averaged stochastic gradient algorithms

Stochastic gradient algorithms are more and more studied since they can deal efficiently and online with large samples in high dimensional spaces. In this paper, we first establish a Central Limit Theorem for these estimates as well as for their averaged version in general Hilbert spaces. Moreover, since having the asymptotic normality of estimates is often unusable without an estimation of the asymptotic variance, we introduce a new recursive algorithm for estimating this last one, and we establish its almost sure rate of convergence as well as its rate of convergence in quadratic mean. Finally, two examples consisting in estimating the parameters of the logistic regression and estimating geometric quantiles are given.

An averaged projected Robbins-Monro algorithm for estimating the parameters of a truncated spherical distribution

The objective of this work is to propose a new algorithm to fit a sphere on a noisy 3D point cloud distributed around a complete or a truncated sphere. More precisely, we introduce a projected Robbins-Monro algorithm and its averaged version for estimating the center and the radius of the sphere. We give asymptotic results such as the almost sure convergence of these algorithms as well as the asymptotic normality of the averaged algorithm. Furthermore, some non-asymptotic results will be given, such as the rates of convergence in quadratic mean. Some numerical experiments show the efficiency of the proposed algorithm on simulated data for small to moderate sample sizes and for modeling an object in 3D.

Estimating the geometric median in Hilbert spaces with stochastic gradient algorithms: Lp and almost sure rates of convergence

The geometric median, also called L 1 -median, is often used in robust statistics. Moreover, it is more and more usual to deal with large samples taking values in high dimensional spaces. In this context, a fast recursive estimator has been introduced by Cardot et al. (2013). This work aims at studying more precisely the asymptotic behavior of the estimators of the geometric median based on such non linear stochastic gradient algorithms. The L p rates of convergence as well as almost sure rates of convergence of these estimators are derived in general separable Hilbert spaces. Moreover, the optimal rates of convergence in quadratic mean of the averaged algorithm are also given.

Wong–Zakai Type Approximations for Stochastic Differential Equations Driven by a Fractional Brownian Motion

We consider Wong–Zakai type approximations for a class of Itô–Volterra equations related to the fractional Brownian motion. The quadratic mean convergence, uniformly on compact time intervals, of the approximations to the solution of an Itô–Volterra equation with a modified drift is obtained.

A strong approximation for double subfractional integrals

A Riemann–Stieltjes integral strong approximation to double Stratonovich integrals with respect to odd and even fractional Brownian motions is considered. We prove the convergence in quadratic mean, uniformly on compact time intervals, of the ordinary double integral process obtained by linear interpolation of the odd and even fractional Brownian motions, to the double Stratonovich integral. The deterministic integrands are continuous or are given by bimeasures.

On the monotone convergence of vector means

Consider a stochastic sequence {Z n; n=1,2,…} , and define P n ( ε)= P(| Z n |< ε). Then the stochastic convergence Z n →0 is said to be monotone whenever the sequence P n ( ε)↑1 monotonically in n for each ε>0. This mode of convergence is investigated here; it is seen to be stronger than convergence in quadratic mean; and scalar and vector sequences exhibiting monotone convergence are demonstrated. In particular, if { X 1,…, X n } is a spherical Cauchy vector whose elements are centered at θ, then Z n =( X 1+⋯+ X n )/ n is not only weakly consistent for θ, but it is shown to follow a monotone law of large numbers. Corresponding results are shown for certain ensembles and mixtures of dependent scalar and vector sequences having n-extendible joint distributions. Supporting facts utilize ordering by majorization; these extend several results from the literature and thus are of independent interest.

Estimating mixing density from a system of observations

Let Y11,…Y1m,Y21…,Yn1,…Ynm be observations such that Yij=Xi+εij, where xi are i.i.d. unobservable random variables with density f and εij are independent measurement errors, with density fε. First, this paper study the estimation of f from the system of observations {Yij}n,m i=1,j=1 when fε is known. Upper bounds on the rate of quadratic mean convergence are obtained. The upper bounds depend on the geometry of the problem and on the rate of growing of m as function of n. Then, we study two special situations inside the system of observations. The first case is when a parameter of the characteristic function of ε is unknown. The second case corresponds to εij non identically distributed. A simulation study has been conducted to show the sampling behavior of the estimators for distinct values of n and m.

Convergence of a quadrature formula for variable-signed weight functions

A quadrature formula for a variable-signed weight function w(x) is constructed using Hermite interpolating polynomials. We show its mean and quadratic mean convergence. We also discuss the rate of convergence in terms of the modulus of continuity for higher order derivatives with respect to the sup norm.

PRODUCT PROCESSES IN VARYING ENVIRONMENTS

A particular type of branching processes in varying environments is considered. It is assumed that all individuals of the same generation produce, given that the preceding generation is not extinct, randomly and independently of the past generations the same number of children. We show that the number of children in the nth generation normed by its expectation converges almost surely to a limit whose expectation is 0 or 1. We give a sufficient condition for convergence in quadratic mean to a limit whose mean is one. A nonclassical norming sequence of constants is defined so that the almost sure limit is finite greater than zero with probability 1. We also show, under certain circumstances, that the almost sure limit has infinite mean.

Spectral analysis of the covariance of the almost periodically correlated processes

This paper deals with the spectrum of the almost periodically correlated (APC) processes defined on R . It is established that the covariance kernel of such a process admits a Fourier series decomposition, K( s+ t, s) ∼∑ α∈ R b α(t)e isα , whose coefficient functions b α are the Fourier transforms of complex measures m α, α∈ R , which are absolutely continuous with respect to the measure m o. Considering the APC strongly harmonizable processes, the spectral covariance of the process can be expressed in terms of these complex measures m α. The usual estimators for the second order situation can be modified to provide consistent estimators of the coefficient functions b α from a sample of the process. Whenever the measures m α are absolutely continuous with respect to the Lebesgue measure, so m α ( dλ)= f α ( λ) dλ, the estimation of the corresponding density functions f α is considered. Under hypotheses on the covariance kernel K and on the coefficient functions b α, we establish rates of convergence in quadratic mean and almost everywhere of these estimators.

Convergence and stability of implicit runge-kutta methods for systems with multiplicative noise

A class ofimplicit Runge-Kutta schemes for stochastic differential equations affected bymultiplicative Gaussian white noise is shown to be optimal with respect to global order of convergence in quadratic mean. A test equation is proposed in order to investigate the stability of discretization methods for systems of this kind. Herestability is intended in a truly probabilistic sense, as opposed to the recently introduced extension of A-stability to the stochastic context, given for systems with additive noise. Stability regions for the optimal class are also given.

On the convergence of iterations disturbed by strong gaussian random operators

The asymptotic behaviour of a sequence of linear iterations distrubed by Gaussian strong random operators is considered. A quadratic mean convergence theorem and analogues of the central limit theorem and of the law of iterated logarithm are proved for the sequence of errors

Density estimation for Markov chains

In this paper kernel estimators for the stationary density of a homogeneous Markov chain with an arbitrary initial distribution are considered under the assumption (D 1) which is weaker than Doeblin’s condition (D 0 ). Condition (D 1) includes periodic Markov chains with cyclic subclasses and an unessential set. Convergence in probability, a.s. convergence and convergence in quadratic mean of these estimators are studied. In this paper estimations for MSE and MISE are given.

A dynamic method of moments

A sequence of univariate or multivariate independent random variables is considered and the problem of estimating their central moments and covariances, is examined. The estimators are based on previous observations and on the assumption that the quantities to be estimated follow unknown trends of a specified form. Convergence in quadratic mean is proved and the corresponding rate of convergence is evaluated. The results obtained are used in order to propose a dymanic method of moments for a time-dependent population.