Obtain Convergence Rates Research Articles

Fullband-subband automatic switching scheme forstereophonicacoustic echo cancellation

A new combination of fullband and subband algorithms, with a particular automatic switching scheme, is proposed to obtain convergence rate improvement and complexity reduction for teleconferencing systems. The switching criterion of the proposed algorithm, based upon the autocorrelation function of the error signal, is introduced to avoid dependency upon signal-to-noise ratio. Simulation examples suggest that enhanced performance can be achieved at low computational cost.

Lois fortes des grands nombres presque sûres pour les sommes de Riesz–Raikov

The classical theorem of Riesz and Raikov states that if a > 1 is an integer and ƒ is a function in L 1(ℝ/ℤ), then the averages converge to the mean value of ƒ over [0, 1] for almost every x in [0, 1]. In this paper we prove that, for ƒ in L 1(ℝ/ℤ), the averages A n a ƒ(x) converge a.e. to the integral of ƒ over [0, 1] for almost every a > 1. Furthermore we obtain convergence rates in this strong law of large numbers.

Numerical solution of integral equations with finite part integrals

We obtain convergence rates for several algorithms that solve a class of Hadamard singular integral equations using the general theory of approximations for unbounded operators.

On the calculation of time-varying stability radii

The problem of calculating the maximal Lyapunov exponent (generalized spectral radius) of a discrete inclusion is formulated as an average yield optimal control problem. It is shown that the maximal value of this problem can be approximated by the maximal value of discounted optimal control problems, where for irreducible inclusions the convergence is linear in the discount rate. This result is used to obtain convergence rates of an algorithm for the calculation of time-varying stability radii. © 1998 John Wiley & Sons, Ltd.

Sieve Extremum Estimates for Weakly Dependent Data

Many non/semi-parametric time series estimates may be regarded as different forms of sieve extremum estimates. For stationary β-mixing observations, we obtain convergence rates of sieve extremum estimates and root-n asymptotic normality of plug-in sieve extremum estimates of smooth functionals. As applications to time series models, we give convergence rates for nonparametric ARX(p,q) regression via neural networks, splines, and wavelets; root-n asymptotic normality for partial linear additive AR(p) models, and monotone transformation AR(1) models.

Some Newton-type methods for the regularization of nonlinear ill-posed problems

In this paper we consider a combination of Newton's method with linear Tikhonov regularization, linear Landweber iteration and truncated SVD, for regularizing an abstract, nonlinear, ill-posed operator equation. We show that under certain smoothness conditions on the nonlinear operator, these methods converge locally. For perturbed data we propose an a priori stopping rule, that guarantees convergence of the iterates to a solution, as the noise level goes to zero. Under appropriate closeness and smoothness assumptions on the starting value and the solution, we obtain convergence rates.

A note on strong convergence rates in nonparametric regression

The strong convergence rates in nonparametric regression estimation have been mostly discussed when the error variables in the regression models have finite variances. A few recent studies concern heavy-tailed error distributions for two comparable methods using the kernel and the k-nearest neighbor estimators. The obtained convergence rates are however noncomparable. Assuming the error variables have finite pth moments for the same p, 1 < p < 2, we derive comparable strong convergence rates for these two estimators via a unified approach. This improves the existing results for both the kernel estimator and the k-nearest neighbor estimator.

Markov Chains for Exploring Posterior Distributions

Several Markov chain methods are available for sampling from a posterior distribution. Two important examples are the Gibbs sampler and the Metropolis algorithm. In addition, several strategies are available for constructing hybrid algorithms. This paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues. On the theoretical side, results from the theory of general state space Markov chains can be used to obtain convergence rates, laws of large numbers and central limit theorems for estimates obtained from Markov chain methods. These theoretical results can be used to guide the construction of more efficient algorithms. For the practical use of Markov chain methods, standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation.

Conditional tail probabilities in continuous-time martingale LLN with application to parameter estimation in diffusions

Let M be a continuous martingale, h: R +→ R + continuous and increasing such that M( t)/ h(F< M> t → 0 (a.s.) as t → ∞. It is shown that w.p.l, large deviations type limits exist for a class of conditional probabilities which are induced on ( C([0, ∞),‖·| ∞) by the tail processes y t (·) = M( t + ·)/ h(< M> t+. ). This is obtained via a simple use of the Borell inequality for Gaussian processes, combined with a random time change argument. Results are applied to obtain convergence rates for the (conditional) tail probabilities of consistent parameter estimators in diffusion processes. This is followed by the derivation of efficient stopping rules. Finally, unconditional large deviations lower bounds for the tails of consistent estimators in diffusions are investigated via an extension of a well known direct method.

On the regularization of projection methods for solving III-posed problems

In this paper we consider a class of regularization methods for a discretized version of an operator equation (which includes the case that the problem is ill-posed) with approximately given right-hand side. We propose an a priori- as well as an a posteriori parameter choice method which is similar to the discrepancy principle of Ivanov-Morozov. From results on fractional powers of selfadjoint operators we obtain convergence rates, which are (in many cases) the same for both parameter choices.

Factorizations that relax the positive real condition in continuous‐time and fast‐sampled ELS schemes

AbstractThis paper proposes extended least‐squares (ELS) for ARMAX model identification of continuous‐time and certain discrete‐time systems. The schemes have a relaxed strictly positive real (SPR) condition for global convergence. The relaxed SPR scheme is achieved by introducing overparametrization and prefiltering but without introducing ill‐conditioning. The schemes presented are the first such proposed for continuous‐time systems.The concepts developed in continuous time carry through to fast‐sampled continuous‐time systems and associated discrete‐time ELS algorithms. For such situations, in comparison with previously proposed discrete‐time schemes, the degree of overparametrization required in the proposed scheme of this paper is significantly lower. The reduction is achieved by using more suitable prefiltering and overparametrization techniques than previously proposed.We also establish the persistence of excitation (PE) of the regression vectors in the proposed ELS schemes to assure strong consistency, obtain convergence rates and provide robustness to unmodelled dynamics. To prove the PE of continuous‐time regression vectors, we develop output reachability characterization for MIMO linear continuous‐time systems.

Factorizations that Relax the Positive Real Condition in Continuous-Time ELS Schemes

This paper proposes Extended Least Squares (ELS) schemes for ARMAX model identification of continuous-time systems. The schemes have a relaxed Strictly Positive Real (SPR) condition for global convergence. The relaxed SPR scheme is achieved by introducing overparametrisation and prefiltering but without introducing ill-conditioning. The schemes presented are the first such proposed for continuous-time systems. The concepts developed here carry through to output-error, fast-sampled continuous-time systems and associated discrete-time ELS algorithms. We also state conditions for the persistence of excitation (P.E.) of the regression vectors in the proposed ELS schemes to assure strong consistency and obtain convergence rates.

A note on the geometric ergodicity of a Markov chain

It is known that if an irreducible and aperiodic Markov chain satisfies a ‘drift' condition in terms of a non-negative measurable function g(x), it is geometrically ergodic. See, e.g. Nummelin (1984), p. 90. We extend the analysis to show that the distance between the nth-step transition probability and the invariant probability measure is bounded above by ρ n (a + bg(x)) for some constants a, b&amp;gt; 0 and ρ &amp;lt; 1. The result is then applied to obtain convergence rates to the invariant probability measures for an autoregressive process and a random walk on a half line.

A note on the geometric ergodicity of a Markov chain

It is known that if an irreducible and aperiodic Markov chain satisfies a ‘drift' condition in terms of a non-negative measurable function g(x), it is geometrically ergodic. See, e.g. Nummelin (1984), p. 90. We extend the analysis to show that the distance between the nth-step transition probability and the invariant probability measure is bounded above by ρ n(a + bg(x)) for some constants a, b&gt; 0 and ρ &lt; 1. The result is then applied to obtain convergence rates to the invariant probability measures for an autoregressive process and a random walk on a half line.

The convergence of several algorithms for solving integral equations with finite part integrals. II

We establish the uniform convergence and obtain convergence rates for several algorithms for solving a class of Hadamard singular integral equations. These results improve on the mean square convergence shown in [2].

A representation for the limiting random variable of a branching process with infinite mean and some related problems

It is known that for a Bienaymé– Galton–Watson process {Zn } whose mean m satisfies 1 &amp;lt; m &amp;lt; ∞, the limiting random variable in the strong limit theorem can be represented as a random sum of i.i.d. random variables and hence that convergence rate results follow from a random sum central limit theorem. This paper develops an analogous theory for the case m = ∞ which replaces ‘sum' by ‘maximum'. In particular we obtain convergence rate results involving a limiting extreme value distribution. An associated estimation problem is considered.

A representation for the limiting random variable of a branching process with infinite mean and some related problems

It is known that for a Bienaymé– Galton–Watson process {Zn} whose mean m satisfies 1 &lt; m &lt; ∞, the limiting random variable in the strong limit theorem can be represented as a random sum of i.i.d. random variables and hence that convergence rate results follow from a random sum central limit theorem.This paper develops an analogous theory for the case m = ∞ which replaces ‘sum' by ‘maximum'. In particular we obtain convergence rate results involving a limiting extreme value distribution. An associated estimation problem is considered.

Variational methods for discontinuous potentials. II. Global trial functions

For pt.I see ibid., vol.9, p.11 (1976). When conventional variational trial functions are used to solve systems such as Schrodinger's equation with discontinuous potentials, it is found that the convergence rate is very slow when compared with that for continuous potentials. In a previous paper, the authors demonstrated how this convergence rate could be improved using local variational methods (i.e. finite elements). In this paper they demonstrate the construction of trial functions for global variational methods which results in a very substantial improvement in the obtained convergence rate. The technique, which is numerically stable, is based on orthogonal polynomials with suitable core functions incorporated.

Convergence Rates for a Class of Large Deviation Probabilities

For a sequence $\{X_n: 1 \leqq n 0$, and $\rho$ is a constant depending on $\lambda$ and the distribution of $X_1$. A 1969 theorem of Sievers, as strengthened by Plachky in 1971, established the convergence of $n^{-1}\log P(W_n \geqq z_n)$ to a constant, where the $W_n$'s have moment-generating functions and belong to a class of random variables more general than partial sums, and the $z_n$'s are numbers such that $n^{-1}z_n \rightarrow \lambda > 0$. In a format related to that of Sievers, Behadur in 1971 analyzed the behavior of $n^{-1}\log P(W_n \geqq z_n)$ in situations when it may not converge to a constant. The goal of the present article is to extend the theorems of Chernoff, Sievers, and Bahadur in the direction of obtaining convergence rates (to 0) of the large deviation probabilities $P(W_n \geqq z_n)$ where the $z_n$'s are numbers such that $n^{-\frac{1}{2}} z_n \rightarrow \infty$. The method of proof is based on the proof of Chernoff's theorem given, in passing, in a 1960 paper of Bahadur and Ranga Rao.