Characteristic Functions and Classical Limit Theorems

2002 ◽  
pp. 83-102
Author(s):  
Olav Kallenberg
Keyword(s):  
Classical Limit ◽  
Limit Theorems ◽  
Characteristic Functions

scholarly journals A Note on Rényi's ‘Record’ Problem and Engel's Series

2018 ◽  
Vol 61 (2) ◽  
pp. 363-369 ◽  
Author(s):  
Lulu Fang ◽  
Min Wu
Keyword(s):  
Number Theory ◽  
Large Deviations ◽  
Markov Processes ◽  
Classical Limit ◽  
Limit Theorems ◽  
London Math ◽  
Law Of Large Numbers ◽  
Record Times ◽  
Iterated Logarithm ◽  
Large Numbers

AbstractIn 1973, Williams [D. Williams, On Rényi's ‘record’ problem and Engel's series, Bull. London Math. Soc.5 (1973), 235–237] introduced two interesting discrete Markov processes, namely C-processes and A-processes, which are related to record times in statistics and Engel's series in number theory respectively. Moreover, he showed that these two processes share the same classical limit theorems, such as the law of large numbers, central limit theorem and law of the iterated logarithm. In this paper, we consider the large deviations for these two Markov processes, which indicate that there is a difference between C-processes and A-processes in the context of large deviations.

Applications of the key renewal theorem: crudely regenerative processes

1992 ◽  
Vol 29 (02) ◽  
pp. 384-395
Author(s):  
Richard F. Serfozo
Keyword(s):  
Classical Limit ◽  
Limit Theorems ◽  
Random Variables ◽  
Limit Function ◽  
Regenerative Processes ◽  
Renewal Theorem ◽  
Key Renewal Theorem ◽  
Derivatives Of

Limit Statements obtainable by the key renewal theorem are of the form EXt = v(t) + o(1), as t →∞. We show how to delineate the limit function v for processes X associated with crudely regenerative phenomena. Included are refinements of classical limit theorems for Markov and regenerative processes, limits of sums of stationary random variables, and limits for integrals and derivatives of EXt.

Non-Classical Limit Theorems

1971 ◽  
Vol 16 (1) ◽  
pp. 175-182 ◽  
Author(s):  
Yu. Yu. Machis
Keyword(s):  
Classical Limit ◽  
Limit Theorems

Applications of the key renewal theorem: crudely regenerative processes

1992 ◽  
Vol 29 (2) ◽  
pp. 384-395 ◽  
Author(s):  
Richard F. Serfozo
Keyword(s):  
Classical Limit ◽  
Limit Theorems ◽  
Random Variables ◽  
Limit Function ◽  
Regenerative Processes ◽  
Renewal Theorem ◽  
Key Renewal Theorem ◽  
Derivatives Of

Limit Statements obtainable by the key renewal theorem are of the form EXt = v(t) + o(1), as t →∞. We show how to delineate the limit function v for processes X associated with crudely regenerative phenomena. Included are refinements of classical limit theorems for Markov and regenerative processes, limits of sums of stationary random variables, and limits for integrals and derivatives of EXt.

A survey of the theory of characteristic functions

1972 ◽  
Vol 4 (01) ◽  
pp. 1-37 ◽  
Author(s):  
Eugene Lukacs
Keyword(s):  
Limit Theorems ◽  
Characteristic Functions ◽  
Main Emphasis ◽  
Recent Developments ◽  
Active Research

The paper gives a survey of the theory of univariate characteristic functions. These functions were originally introduced as tools in the study of limit theorems but it was later realized that they had an independent mathematical interest. Those parts of the theory which can be found in textbooks are treated only briefly; the main emphasis is placed on more recent developments and areas where active research is still in progress.

The Classical limit of the space distribution law of a gas in a field of force

1928 ◽  
Vol 24 (1) ◽  
pp. 76-79 ◽  
Author(s):  
N. F. Mott
Keyword(s):  
Quantum Theory ◽  
Statistical Mechanics ◽  
Classical Theory ◽  
Classical Limit ◽  
Space Distribution ◽  
Characteristic Functions ◽  
Distribution Law ◽  
Distribution Laws ◽  
The Law ◽  
Analogous Method

The Statistical Mechanics which has been developed in accordance with the requirements of the new Quantum Theory is concerned with distribution laws over energy values only—over, that is, the characteristics of Schrödinger's equation. To obtain a space distribution law, even for the Classical limit, some use must be made of the characteristic functions. A formula has been suggested by Fowler, but it has not been shown that this formula gives the Classical law for gases at ordinary temperatures and pressures. In this paper we shall show that this is so, but before doing so we shall sketch the analogous method of obtaining the law, on the Classical theory.

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