The rate of convergence

In this study, a random walk process with generalized reflecting barrier is considered and an inequality for rate of weak convergence of the stationary distribution of the process of interest is propounded. Though the rate of convergence is not thoroughly examined, the literature does provide a weak convergence theorem under certain conditions for the stationary distribution of the process under consideration. Nonetheless, one of the most crucial issues in probability theory is the convergence rate in limit theorems, as it affects the precision and effectiveness of using these theorems in practice. Therefore, for the rate of convergence for the examined process, comparatively simple inequality is represented. The obtained inequality demonstrates that the rate of convergence is correlated with the tail of the distribution of ladder heights of the random walk.

On the Rate of Convergence for the Invariance Principle

On the Chebyshev-Cramér Asymptotic Expansions

A Non-Uniform Estimate for the Convergence Speed in the Multi-Dimensional Central Theorem

By using the theory of the second-order regular variation, we study the rates of the weak convergence of the maximum order statistics under power normalization. The exact rates are obtained in the uniform metric and the total variation metric. The relationship between the rates of convergence under linear and under power normalization is derived. Some illustrative examples are given for comparing the rates of convergence.

For a stochastic differential equation(SDE) driven by a fractional Brownian motion(fBm) with Hurst parameter H>12, it is known that the existing (naive) Euler scheme has the rate of convergence n1−2H. Since the limit H→12 of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler scheme is the extension of the classical Euler scheme for Itô SDEs for H=12, the convergence rate of the naive Euler scheme deteriorates for H→12. In this paper we introduce a new (modified Euler) approximation scheme which is closer to the classical Euler scheme for Stratonovich SDEs for H=12, and it has the rate of convergence γ−1n, where γn=n2H−1/2 when H<34, γn=n/logn−−−−√ when H=34 and γn=n if H>34. Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if {Xt,0≤t≤T} is the solution of a SDE driven by a fBm and if {Xnt,0≤t≤T} is its approximation obtained by the new modified Euler scheme, then we prove that γn(Xn−X) converges stably to the solution of a linear SDE driven by a matrix-valued Brownian motion, when H∈(12,34]. In the case H>34, we show the Lp convergence of n(Xnt−Xt), and the limiting process is identified as the solution of a linear SDE driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this scheme. We also apply our approach to the naive Euler scheme.

In this paper we study the rate of local convergence of the augmented Lagrangian method for the semidefinite nuclear norm composite optimization problem. We propose two basic assumptions, under which the convergence rate of the augmented Lagrange method for a general composite optimization problems is estimated. We analyze the rate of local convergence of the augmented Lagrangian method for the nonlinear semidefinite nuclear norm composite optimization problem by verifying these two basic assumptions. Without requiring strict complementarity, we prove that, under the constraint nondegeneracy condition and the strong second order sufficient condition, the rate of convergence is linear and the ratio constant is proportional to 1/c, where c is the penalty parameter that exceeds a threshold c ¯ > 0 . The analysis is based on variational analysis about the proximal mapping of the nuclear norm and the projection operator onto the cone of positively semidefinite symmetric matrices.

On the Accuracy of Gaussian Approximation to the Distribution Functions of Sums of Independent Variables

Statistical learning theory on probability space is an important part of machine learning. Based on the key theorem, the bounds on the rate of relative uniform convergence have significant meaning. These bounds determine generalization ability of the learning machines utilizing the empirical risk minimization induction principle. In this paper, the bounds of the learning processes on possibility space are discussed, and the rate of relative uniform convergence is estimated, and finally the relation between the rate of convergence and the capacity of a set of function is pointed out.

Using a representation formula expressing the mixed cumulants of realvalued random variables by corresponding moments, sufficient conditions are given for the normal convergence of suitably standardized shot noise assuming that the generating stationary point process is independently marked and Brillinger mixing and that its intensity tends to ∞. Furthermore, estimates for the rate of this normal convergence are obtained by exploiting a general lemma on probabilities of large deviations and on the rate of normal convergence.

We study the local convergence of classical quasi-Newton methods for nonlinear optimization. Although it was well established a long time ago that asymptotically these methods converge superlinearly, the corresponding rates of convergence still remain unknown. In this paper, we address this problem. We obtain first explicit non-asymptotic rates of superlinear convergence for the standard quasi-Newton methods, which are based on the updating formulas from the convex Broyden class. In particular, for the well-known DFP and BFGS methods, we obtain the rates of the form (frac{n L^2}{mu ^2 k})^{k/2} and (frac{n L}{mu k})^{k/2} respectively, where k is the iteration counter, n is the dimension of the problem, mu is the strong convexity parameter, and L is the Lipschitz constant of the gradient.

Using a representation formula expressing the mixed cumulants of realvalued random variables by corresponding moments, sufficient conditions are given for the normal convergence of suitably standardized shot noise assuming that the generating stationary point process is independently marked and Brillinger mixing and that its intensity tends to ∞. Furthermore, estimates for the rate of this normal convergence are obtained by exploiting a general lemma on probabilities of large deviations and on the rate of normal convergence.

Uniform Estimates of the Rate of Convergence in the Multi-Dimensional Central Limit Theorem

This note addresses the following question. Suppose we have a sequence of distributions of sums of independent identically distributed (i.i.d.) random summands converging to a given stable law. What properties of the summands have the most influence on the rate of convergence under consideration? At once it is necessary to note that a general answer is well known at present (a good reference for the accuracy of approximation with stable laws is the recent monograph [3], see also [6]-[11]), and it can be roughly stated that the existence of pseudomoments of higher order ensures better rates of convergence. In turn, the existence of pseudomoments depends on how close are the distribution of the summand and the stable law. But contrary to the case of the Gaussian limit law, where the existence of moments (or more precisely, tail behavior of the distribution) defines the rate of convergence and necessary and sufficient conditions for a given rate of convergence can be formulated, there is no such natural measure of closeness of the distributions in the case of the stable limit law. Taking the particular form of the density of the summand, namely, the Paretian distribution, which can be regarded in some sense as the "main" distribution in the domain of normal attraction (DNA) of a given stable distribution, we show how the rate of convergence depends on small perturbations of this density. Another goal of this note was to give an answer to the question raised in [1]. In that paper it was noted that the convergence to a stable distribution may be extremely slow (this was shown by means of simulation). It turns out that in the situation described in that paper summands belong to the domain of attraction (DA) but do not belong to the DNA. Our example shows that in this case the rate of convergence is only of logarithmic rate. This means that although theoretically both DNA and DA are equally important, in practice, limit theorems in the case of summands in DA but not in DNA are less useful due to the extremely slow rate of convergence and the fact that even asymptotic expansions cannot help very much (see Propositions 2 and 3, where new types of asymptotic expansions are given).

Intelligent control of convergence rate of impulsive dynamic systems affected by nonlinear disturbances under stabilizing impulses and its application in Chua’s circuit