scholarly journals Convergence in capacity of rational approximants of meromorphic functions

2014 ◽  
Vol 96 (110) ◽  
pp. 31-39 ◽  
Author(s):  
Hans-Peter Blatt
Keyword(s):  
Convergence Rate ◽  
Meromorphic Functions ◽  
Compact Set ◽  
Rational Approximants ◽  
Geometric Convergence ◽  
Convergence In Capacity

Let f be meromorphic on the compact set E ? C with maximal Green domain of meromorphy Ep(f), p(f) < ?. We investigate rational approximants with numerator degree ? n and denominator degree ? mn for f. We show that the geometric convergence rate on E implies convergence in capacity outside E if mn = o(n) as n ? 1. Further, we show that the condition is sharp and that the convergence in capacity is uniform for a subsequence ? ? N.

scholarly journals Convergence rates for estimators of geodesic distances and Frèchet expectations

2018 ◽  
Vol 55 (4) ◽  
pp. 1001-1013
Author(s):  
Catherine Aaron ◽  
Olivier Bodart
Keyword(s):  
Probability Distribution ◽  
Convergence Rate ◽  
Compact Manifold ◽  
Convergence Rates ◽  
Geodesic Distance ◽  
Convergence Result ◽  
Regularity Conditions ◽  
Compact Set ◽  
Geodesic Distances ◽  
General Convergence

Abstract Consider a sample 𝒳n={X1,…,Xn} of independent and identically distributed variables drawn with a probability distribution ℙX supported on a compact set M⊂ℝd. In this paper we mainly deal with the study of a natural estimator for the geodesic distance on M. Under rather general geometric assumptions on M, we prove a general convergence result. Assuming M to be a compact manifold of known dimension d′≤d, and under regularity assumptions on ℙX, we give an explicit convergence rate. In the case when M has no boundary, knowledge of the dimension d′ is not needed to obtain this convergence rate. The second part of the work consists in building an estimator for the Fréchet expectations on M, and proving its convergence under regularity conditions, applying the previous results.

Geometric convergence of value-iteration in multichain Markov decision problems

1979 ◽  
Vol 11 (01) ◽  
pp. 188-217 ◽  
Author(s):  
P. J. Schweitzer ◽  
A. Federgruen
Keyword(s):  
Convergence Rate ◽  
Iteration Method ◽  
Chain Structure ◽  
Decision Problems ◽  
Value Iteration ◽  
Convergence Factor ◽  
Markov Decision Problems ◽  
Geometric Convergence ◽  
Markov Decision ◽  
Maximal Gain

This paper considers undiscounted Markov decision problems. With no restriction (on either the periodicity or chain structure of the problem) we show that the value iteration method for finding maximal gain policies exhibits a geometric rate of convergence, whenever convergence occurs. In addition, we study the behaviour of the value-iteration operator; we give bounds for the number of steps needed for contraction, describe the ultimate behaviour of the convergence factor and give conditions for the existence of a uniform convergence rate.

Rational Interpolation of the Exponential Function

1995 ◽  
Vol 47 (6) ◽  
pp. 1121-1147 ◽  
Author(s):  
L. Baratchart ◽  
E. B. Saff ◽  
F. Wielonsky
Keyword(s):  
Rational Function ◽  
Exponential Function ◽  
Complex Plane ◽  
Rational Interpolation ◽  
Real Interval ◽  
Compact Set ◽  
Rational Approximants ◽  
Sharp Estimates ◽  
Image Position ◽  
Uniformly Bounded

AbstractLet m, n be nonnegative integers and B(m+n) be a set of m + n + 1 real interpolation points (not necessarily distinct). Let Rm,n = Pm,n/Qm.n be the unique rational function with deg Pm,n ≤ m, deg Qm,n ≤ n, that interpolates ex in the points of B(m+n). If m = mv, n = nv with mv + nv → ∞, and mv / nv → λ as v → ∞, and the sets B(m+n) are uniformly bounded, we show that locally uniformly in the complex plane C, where the normalization Qm,n(0) = 1 has been imposed. Moreover, for any compact set K ⊂ C we obtain sharp estimates for the error |ez — Rm,n(z)| when z ∈ K. These results generalize properties of the classical Padé approximants. Our convergence theorems also apply to best (real) Lp rational approximants to ex on a finite real interval.

Geometric renewal convergence rates from hazard rates

2001 ◽  
Vol 38 (01) ◽  
pp. 180-194 ◽  
Author(s):  
Kenneth S. Berenhaut ◽  
Robert Lund
Keyword(s):  
Convergence Rate ◽  
Hazard Rate ◽  
Convergence Rates ◽  
Hazard Rates ◽  
Finite Support ◽  
Good Convergence ◽  
Renewal Sequence ◽  
Geometric Convergence ◽  
New Better Than Used ◽  
Better Than

This paper studies the geometric convergence rate of a discrete renewal sequence to its limit. A general convergence rate is first derived from the hazard rates of the renewal lifetimes. This result is used to extract a good convergence rate when the lifetimes are ordered in the sense of new better than used or increasing hazard rate. A bound for the best possible geometric convergence rate is derived for lifetimes having a finite support. Examples demonstrating the utility and sharpness of the results are presented. Several of the examples study convergence rates for Markov chains.

A comparison of the ordinary and a varying parameter exponential smoothing

1989 ◽  
Vol 26 (4) ◽  
pp. 784-792 ◽  
Author(s):  
Heikki Bonsdorff
Keyword(s):  
Convergence Rate ◽  
Markov Chains ◽  
Finite Interval ◽  
Exponential Smoothing ◽  
Uniform Ergodicity ◽  
Geometric Convergence ◽  
Monotonically Increasing Function ◽  
Increasing Function

An adaptive-type exponential smoothing, motivated by an insurance tariff problem, is treated. We consider the process Zn = ß(Zn –1)Xn +(1 – ß (Zn–1))Zn–1, where Xn are i.i.d. taking values in the interval [0, M], M ≦ ∞ and ß is a monotonically increasing function [0, M] → [c, d], 0 < c < d < 1.Together with (Zn), we consider the ordinary exponential smoothing Yn = αXn + (1 – α)Yn –1 where α is a constant, 0 < α < 1. We show that (Yn) and (Zn) are geometrically ergodic Markov chains (in the case of finite interval we even have uniform ergodicity) and that EYn, EZn converge to limits EY, EZ, respectively, with a geometric convergence rate. Moreover, we show that Ez is strictly less than EY = EXn.

scholarly journals Approximation by Certain Linear Positive Operators of Two Variables

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Afşin Kürşat Gazanfer ◽  
İbrahim Büyükyazıcı
Keyword(s):  
Convergence Rate ◽  
Modulus Of Continuity ◽  
Weighted Approximation ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Compact Set ◽  
Approximation Properties ◽  
Linear Positive Operators ◽  
Functions Of Two Variables

We introduce positive linear operators which are combined with the Chlodowsky and Szász type operators and study some approximation properties of these operators in the space of continuous functions of two variables on a compact set. The convergence rate of these operators are obtained by means of the modulus of continuity. And we also obtain weighted approximation properties for these positive linear operators in a weighted space of functions of two variables and find the convergence rate for these operators by using the weighted modulus of continuity.

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