scholarly journals Uniform decomposition of probability measures: quantization, clustering and rate of convergence

2018 ◽  
Vol 55 (4) ◽  
pp. 1037-1045 ◽  
Author(s):  
Julien Chevallier
Keyword(s):  
Rate Of Convergence ◽  
Uniform Approximation ◽  
Approximation Error ◽  
Upper Bounds ◽  
Probability Measures ◽  
Finite Approximations ◽  
Elementary Construction

AbstractThe study of finite approximations of probability measures has a long history. In Xu and Berger (2017), the authors focused on constrained finite approximations and, in particular, uniform ones in dimensiond=1. In the present paper we give an elementary construction of a uniform decomposition of probability measures in dimensiond≥1. We then use this decomposition to obtain upper bounds on the rate of convergence of the optimal uniform approximation error. These bounds appear to be the generalization of the ones obtained by Xu and Berger (2017) and to be sharp for generic probability measures.

scholarly journals Least upper bounds for probability measures and their applications to abstractions

2014 ◽  
Vol 234 ◽  
pp. 68-106
Author(s):  
Rohit Chadha ◽  
Mahesh Viswanathan ◽  
Ramesh Viswanathan
Keyword(s):  
Upper Bounds ◽  
Probability Measures

scholarly journals Bounds on the rate of convergence for one class of inhomogeneous Markovian queueing models with possible batch arrivals and services

2018 ◽  
Vol 28 (1) ◽  
pp. 141-154 ◽  
Author(s):  
Alexander Zeifman ◽  
Rostislav Razumchik ◽  
Yacov Satin ◽  
Ksenia Kiseleva ◽  
Anna Korotysheva ◽  
...  
Keyword(s):  
Rate Of Convergence ◽  
Queueing Systems ◽  
Approximation Error ◽  
Linear Operators ◽  
Upper And Lower Bounds ◽  
Logarithmic Norm ◽  
Batch Arrivals ◽  
State Dependent ◽  
The Mean ◽  
Number Of Customers

AbstractIn this paper we present a method for the computation of convergence bounds for four classes of multiserver queueing systems, described by inhomogeneous Markov chains. Specifically, we consider an inhomogeneous M/M/S queueing system with possible state-dependent arrival and service intensities, and additionally possible batch arrivals and batch service. A unified approach based on a logarithmic norm of linear operators for obtaining sharp upper and lower bounds on the rate of convergence and corresponding sharp perturbation bounds is described. As a side effect, we show, by virtue of numerical examples, that the approach based on a logarithmic norm can also be used to approximate limiting characteristics (the idle probability and the mean number of customers in the system) of the systems considered with a given approximation error.

Mixing rates for Brownian motion in a convex polyhedron

1990 ◽  
Vol 27 (2) ◽  
pp. 259-268 ◽  
Author(s):  
Peter Matthews
Keyword(s):  
Brownian Motion ◽  
Rate Of Convergence ◽  
Convex Polyhedron ◽  
Upper Bounds ◽  
Convergence In Distribution ◽  
Convex Polyhedral ◽  
Variation Distance ◽  
Mixing Rates

For Brownian motion on a convex polyhedral subset of a sphere or torus, the rate of convergence in distribution to uniformity is studied. The main result is a method to take a Markov coupling on the full sphere or torus and create a faster coupling on the convex polyhedral subset. Upper bounds on variation distance are computed, and applications are discussed.

Upper bounds on the rate of convergence for constant retrial rate queueing model with two servers

2018 ◽  
Vol 59 (4) ◽  
pp. 1271-1282
Author(s):  
Yacov Satin ◽  
Evsey Morozov ◽  
Ruslana Nekrasova ◽  
Alexander Zeifman ◽  
Ksenia Kiseleva ◽  
...  
Keyword(s):  
Rate Of Convergence ◽  
Upper Bounds ◽  
Queueing Model ◽  
Constant Retrial Rate

scholarly journals Equidistribution results for geodesic flows

2013 ◽  
Vol 34 (3) ◽  
pp. 742-764
Author(s):  
ABDELHAMID AMROUN
Keyword(s):  
Geodesic Flow ◽  
Large Deviation ◽  
Riemannian Metric ◽  
Upper Bounds ◽  
Equilibrium States ◽  
Topological Pressure ◽  
Probability Measures ◽  
Geodesic Flows ◽  
Lower And Upper Bounds ◽  
Unit Tangent Bundle

AbstractUsing the works of Mañé [On the topological entropy of the geodesic flows.J. Differential Geom.45(1989), 74–93] and Paternain [Topological pressure for geodesic flows.Ann. Sci. Éc. Norm. Supér.(4)33(2000), 121–138] we study the distribution of geodesic arcs with respect to equilibrium states of the geodesic flow on a closed manifold, equipped with a$\mathcal {C}^{\infty }$Riemannian metric. We prove large-deviation lower and upper bounds and a contraction principle for the geodesic flow in the space of probability measures of the unit tangent bundle. We deduce a way of approximating equilibrium states for continuous potentials.

Bounds for the total variation distance between the binomial and the Poisson distribution in case of medium-sized success probabilities

1999 ◽  
Vol 36 (01) ◽  
pp. 97-104 ◽  
Author(s):  
Michael Weba
Keyword(s):  
Total Variation ◽  
Poisson Distribution ◽  
Approximation Error ◽  
Upper Bounds ◽  
Risk Theory ◽  
Total Variation Distance ◽  
Medium Size ◽  
Bernoulli Random Variables ◽  
Variation Distance ◽  
Approximation Problems

In applied probability, the distribution of a sum of n independent Bernoulli random variables with success probabilities p 1,p 2,…, p n is often approximated by a Poisson distribution with parameter λ = p 1 + p 2 + p n . Popular bounds for the approximation error are excellent for small values, but less efficient for moderate values of p 1,p 2,…,p n . Upper bounds for the total variation distance are established, improving conventional estimates if the success probabilities are of medium size. The results may be applied directly, e.g. to approximation problems in risk theory.

scholarly journals An Integral Upper Bound for Neural Network Approximation

2009 ◽  
Vol 21 (10) ◽  
pp. 2970-2989 ◽  
Author(s):  
Paul C. Kainen ◽  
Věra Kůrková
Keyword(s):  
Neural Network ◽  
Upper Bound ◽  
Nonlinear Approximation ◽  
Approximation Error ◽  
Upper Bounds ◽  
Integral Representations ◽  
Integration Theory ◽  
Hidden Layer

Complexity of one-hidden-layer networks is studied using tools from nonlinear approximation and integration theory. For functions with suitable integral representations in the form of networks with infinitely many hidden units, upper bounds are derived on the speed of decrease of approximation error as the number of network units increases. These bounds are obtained for various norms using the framework of Bochner integration. Results are applied to perceptron networks.

scholarly journals Bounds on the Rate of Convergence for MtX/MtX/1 Queueing Models

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1752
Author(s):  
Alexander Zeifman ◽  
Yacov Satin ◽  
Alexander Sipin
Keyword(s):  
Markov Chains ◽  
Rate Of Convergence ◽  
Continuous Time ◽  
Upper Bounds ◽  
Specific Class ◽  
Differential Inequalities ◽  
Method Of Differential Inequalities ◽  
Finite Markov Chains ◽  
Intensity Functions ◽  
Forward System

We apply the method of differential inequalities for the computation of upper bounds for the rate of convergence to the limiting regime for one specific class of (in)homogeneous continuous-time Markov chains. Such an approach seems very general; the corresponding description and bounds were considered earlier for finite Markov chains with analytical in time intensity functions. Now we generalize this method to locally integrable intensity functions. Special attention is paid to the situation of a countable Markov chain. To obtain these estimates, we investigate the corresponding forward system of Kolmogorov differential equations as a differential equation in the space of sequences l1.

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