Functions of discrete probability measures: Rates of convergence in the renewal theorem

1983 ◽  
Vol 64 (3) ◽  
pp. 341-357 ◽  
Author(s):  
Rudolf Grübel
Keyword(s):  
Rates Of Convergence ◽  
Probability Measures ◽  
Renewal Theorem ◽  
Discrete Probability

Limiting properties of Inverse Beta and generalized Bleimann–Butzer–Hahn operators

1993 ◽  
Vol 114 (3) ◽  
pp. 489-498 ◽  
Author(s):  
José A. Adell ◽  
Jesús De La Cal
Keyword(s):  
Total Variation ◽  
Rates Of Convergence ◽  
Linear Operators ◽  
Total Variation Distance ◽  
Probability Measures ◽  
Baskakov Operators ◽  
Variation Distance ◽  
Probabilistic Type

AbstractIn this paper, we consider limiting properties concerning linear operators of probabilistic type. Specifically, we show that gamma operators are limits of inverse Beta operators and that Bleimann–Butzer–Hahn, Szász and Baskakov operators are limits of generalized Bleimann–Butzer–Hahn operators. By duality, these results are closely related to the convergence in the total variation distance of the probability measures involved. In each case, rates of convergence are given.

Weak convergence in queueing theory

1973 ◽  
Vol 5 (3) ◽  
pp. 570-594 ◽  
Author(s):  
Donald L. Iglehart
Keyword(s):  
Weak Convergence ◽  
Queueing Theory ◽  
Heavy Traffic ◽  
Rates Of Convergence ◽  
Probability Measures ◽  
Queueing Models ◽  
Applied Probability ◽  
Light Traffic ◽  
Survey Paper ◽  
Convergence Of Probability Measures

In the last ten years the theory of weak convergence of probability measures has been used extensively in studying the models of applied probability. By far the greatest consumer of weak convergence has been the area of queueing theory. This survey paper represents an attempt to summarize the experience in queueing theory with the hope that it will prove helpful in other areas of applied probability. The paper is organized into the following sections: queues in light traffic, queues in heavy traffic, queues with a large number of servers, continuity of queues, rates of convergence, and special queueing models.

scholarly journals Some results of multidimensional discrete probability measures represented by Euler products

JSIAM Letters ◽  
2014 ◽  
Vol 6 (0) ◽  
pp. 45-48
Author(s):  
Takahiro Aoyama ◽  
Nobutaka Shimizu
Keyword(s):  
Probability Measures ◽  
Discrete Probability ◽  
Euler Products

On the unimodality of discrete probability measures

1989 ◽  
Vol 201 (1) ◽  
pp. 131-137 ◽  
Author(s):  
Emile Bertin ◽  
Radu Theodorescu
Keyword(s):  
Probability Measures ◽  
Discrete Probability

scholarly journals Relative and Discrete Utility Maximising Entropy

2008 ◽  
Vol 15 (04) ◽  
pp. 303-327
Author(s):  
Grzegorz Harańczyk ◽  
Wojciech Słomczyński ◽  
Tomasz Zastawniak
Keyword(s):  
Probability Measure ◽  
Utility Function ◽  
Expected Utility ◽  
Mathematical Finance ◽  
Second Law ◽  
Probability Measures ◽  
Discrete Probability ◽  
Terminal Wealth ◽  
Probability Spaces ◽  
Rényi Relative Entropy

The notion of utility maximising entropy (u-entropy) of a probability density, which was introduced and studied in [37], is extended in two directions. First, the relative u-entropy of two probability measures in arbitrary probability spaces is defined. Then, specialising to discrete probability spaces, we also introduce the absolute u-entropy of a probability measure. Both notions are based on the idea, borrowed from mathematical finance, of maximising the expected utility of the terminal wealth of an investor. Moreover, u-entropy is also relevant in thermodynamics, as it can replace the standard Boltzmann-Shannon entropy in the Second Law. If the utility function is logarithmic or isoelastic (a power function), then the well-known notions of Boltzmann-Shannon and Rényi relative entropy are recovered. We establish the principal properties of relative and discrete u-entropy and discuss the links with several related approaches in the literature.

Weak convergence in queueing theory

1973 ◽  
Vol 5 (03) ◽  
pp. 570-594 ◽  
Author(s):  
Donald L. Iglehart
Keyword(s):  
Weak Convergence ◽  
Queueing Theory ◽  
Heavy Traffic ◽  
Rates Of Convergence ◽  
Probability Measures ◽  
Queueing Models ◽  
Applied Probability ◽  
Light Traffic ◽  
Survey Paper ◽  
Convergence Of Probability Measures

In the last ten years the theory of weak convergence of probability measures has been used extensively in studying the models of applied probability. By far the greatest consumer of weak convergence has been the area of queueing theory. This survey paper represents an attempt to summarize the experience in queueing theory with the hope that it will prove helpful in other areas of applied probability. The paper is organized into the following sections: queues in light traffic, queues in heavy traffic, queues with a large number of servers, continuity of queues, rates of convergence, and special queueing models.

scholarly journals Optimal Quantization for Piecewise Uniform Distributions

2018 ◽  
Vol 13 (2) ◽  
pp. 23-55 ◽  
Author(s):  
Joseph Rosenblatt ◽  
Mrinal Kanti Roychowdhury
Keyword(s):  
Finite Number ◽  
Uniform Distribution ◽  
Infinite Number ◽  
Independent Random Variables ◽  
Probability Measures ◽  
Discrete Probability ◽  
Uniform Distributions ◽  
Optimal Quantization ◽  
Positive Integers ◽  
Optimal Set

Abstract Quantization for a probability distribution refers to the idea of estimating a given probability by a discrete probability supported by a finite number of points. In this paper, firstly a general approach to this process is outlined using independent random variables and ergodic maps; these give asymptotically the optimal sets of n-means and the nth quantization errors for all positive integers n. Secondly two piecewise uniform distributions are considered on R: one with infinite number of pieces and one with finite number of pieces. For these two probability measures, we describe the optimal sets of n-means and the nth quantization errors for all n ∈ N. It is seen that for a uniform distribution with infinite number of pieces to determine the optimal sets of n-means for n ≥ 2 one needs to know an optimal set of (n − 1)-means, but for a uniform distribution with finite number of pieces one can directly determine the optimal sets of n-means and the nth quantization errors for all n ∈ N.

Sign in / Sign up

Export Citation Format

Share Document