scholarly journals Applications of subsequential Tauberian theory to classical Tauberian theory

2007 ◽  
Vol 20 (8) ◽  
pp. 946-950 ◽  
Author(s):  
Filiz Dik ◽  
Mehmet Dik ◽  
İbrahim Çanak
Keyword(s):  
Tauberian Theory

scholarly journals Sir Harry Raymond Pitt. 3 June 1914 — 8 October 2005

2008 ◽  
Vol 54 ◽  
pp. 257-274 ◽  
Author(s):  
N. H. Bingham ◽  
W. K. Hayman
Keyword(s):  
Tauberian Theorem ◽  
Norbert Wiener ◽  
Vice Chancellor ◽  
Tauberian Theory

Sir Harry Pitt worked (as H. R. Pitt) with Norbert Wiener in 1938 on Wiener's general Tauberian theory. Mathematically, he is best known for Pitt's form of Wiener's Tauberian theorem, and as the author of the first (1958) monograph on Tauberian theory. He is otherwise best known for having been Vice–Chancellor of Reading University from 1964 to 1978.

Tauberian theory for the asymptotic forms of statistical frequency functions

1952 ◽  
Vol 48 (4) ◽  
pp. 592-599 ◽  
Author(s):  
J. M. Hammersley
Keyword(s):  
Tauberian Theorem ◽  
Wide Sense ◽  
Individual Values ◽  
Tauberian Condition ◽  
Tauberian Theory ◽  
Abelian Theorem ◽  
Statistical Frequency ◽  
Limit Behaviour ◽  
The Individual ◽  
Image Position

An Abelian theorem is a theorem stating that a given behaviour on the part of each of several quantities entails similar behaviour for their average. A Tauberian theorem is a converse to an Abelian theorem. As a rule, a given behaviour of an average will not entail similar behaviour of the individual quantities themselves unless there is some condition imposed to secure reasonably uniform behaviour amongst the individuals. Such a condition, known as a Tauberian condition, is usually sufficient but not necessary, and it enters into the premises of the Tauberian theorem. We interpret ‘average’ in a wide sense to include any kind of smoothing process; for example, the integral of a function f(t) is an average of the values of f(t) corresponding to individual values of t; and we may seek a sufficient Tauberian condition such that a limit-behaviour of an integral-averageentails the corresponding limit-behaviourfor individual values of t.

scholarly journals Reliability Analysis of a Cold Standby System with Imperfect Repair and under Poisson Shocks

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Yutian Chen ◽  
Xianyun Meng ◽  
Shengqiang Chen
Keyword(s):  
Poisson Process ◽  
Reliability Analysis ◽  
System Availability ◽  
Reliability Indices ◽  
Intrinsic Factors ◽  
Cold Standby ◽  
Tauberian Theory ◽  
The Mean ◽  
Standby System ◽  
Mean Time

This paper considers the reliability analysis of a two-component cold standby system with a repairman who may have vacation. The system may fail due to intrinsic factors like aging or deteriorating, or external factors such as Poisson shocks. The arrival time of the shocks follows a Poisson process with the intensityλ>0. Whenever the magnitude of a shock is larger than the prespecified threshold of the operating component, the operating component will fail. The paper assumes that the intrinsic lifetime and the repair time on the component are an extended Poisson process, the magnitude of the shock and the threshold of the operating component are nonnegative random variables, and the vacation time of the repairman obeys the general continuous probability distribution. By using the vector Markov process theory, the supplementary variable method, Laplace transform, and Tauberian theory, the paper derives a number of reliability indices: system availability, system reliability, the rate of occurrence of the system failure, and the mean time to the first failure of the system. Finally, a numerical example is given to validate the derived indices.

On the limit of a supercritical branching process

1988 ◽  
Vol 25 (A) ◽  
pp. 215-228 ◽  
Author(s):  
N. H. Bingham
Keyword(s):  
Continuous Time ◽  
Branching Process ◽  
Recent Progress ◽  
Branching Processes ◽  
Random Variable ◽  
Tail Behaviour ◽  
Tauberian Theory ◽  
The Right ◽  
Schröder Case ◽  
Böttcher Case

LetWbe the usual almost-sure limit random variable in a supercritical simple branching process; we study its tail behaviour. For the left tail, we distinguish two cases, the ‘Schröder' and ‘Böttcher' cases; both appear in work of Harris and Dubuc. The Schröder case is related to work of Karlin and McGregor on embeddability in continuous-time (Markov) branching processes. New results are obtained for the Böttcher case; there are links with recent work of Barlow and Perkins on Brownian motion on a fractal. The right tail is also considered. Use is made of recent progress in Tauberian theory.

On the limit of a supercritical branching process

1988 ◽  
Vol 25 (A) ◽  
pp. 215-228 ◽  
Author(s):  
N. H. Bingham
Keyword(s):  
Continuous Time ◽  
Branching Process ◽  
Recent Progress ◽  
Branching Processes ◽  
Random Variable ◽  
Tail Behaviour ◽  
Tauberian Theory ◽  
The Right ◽  
Schröder Case ◽  
Böttcher Case

Let W be the usual almost-sure limit random variable in a supercritical simple branching process; we study its tail behaviour. For the left tail, we distinguish two cases, the ‘Schröder' and ‘Böttcher' cases; both appear in work of Harris and Dubuc. The Schröder case is related to work of Karlin and McGregor on embeddability in continuous-time (Markov) branching processes. New results are obtained for the Böttcher case; there are links with recent work of Barlow and Perkins on Brownian motion on a fractal. The right tail is also considered. Use is made of recent progress in Tauberian theory.

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