Lower Estimates of the Convergence Rate for $U$-Statistics

Asymptotics for the Moment Convergence of U-Statistics in LIL

Weak Convergence of Generalized U - Statistics

Convergence of U-statistics for interacting particle systems

The convergence of U -statistics has been intensively studied for estimators based on families of i.i.d. random variables and variants of them. In most cases, the independence assumption is crucial [Lee90, dlPG99]. When dealing with Feynman-Kac and other interacting particle systems of Monte Carlo type, one faces a new type of problem. Namely, in a sample of N particles obtained through the cor...

Improvement of the rate of convergence estimates for multigrid algorithm

In this paper, we present a new convergence rate for both symmetric and nonsymmetric positive definite problems. In our theory, ‘‘regularity and approximation’’ assumption is used. A new rate of convergence estimate with a 1⁄4 1 2 is used where a is a ‘‘regularity and approximation’’ parameter. Also, a new convergence rate is given by two-grid schemes. 2006 Elsevier Inc. All rights reserved.

Convergence rate estimates for the gradient differential inclusion

Let f : H → R ∪ {∞} be a proper, lower semi–continuous, convex function in a Hilbert space H. The gradient differential inclusion is x′(t) ∈ −∂f(x(t)), x(0) = x, where x ∈ dom(f). If f is differentiable, the inclusion can be considered as the continuous version of the steepest descent method for minimizing f on H. Even if f is not differentiable, the inclusion has a unique solution {x(t) : t > ...