Large Deviation Local Limit Theorems for Ratio Statistics
Large Deviation Local Limit Theorems for Ratio Statistics

Author: Narasinga Rao Chaganty

Publisher:

Published: 1987

Total Pages: 204

ISBN-13:

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This document discusses an arbitrary sequence of non-lattice random variables and another sequence of positive random variables. Assume that the sequences are independent. This paper obtains asymptotic expression for the density function of the ratio statistic R sub n = T sub n/S sub n based on simple conditions on the moment generating functions of T sub n and S sub n. When S sub n = n, our main result reduces to that of Chaganty and Sethuraman. We also obtain analogous results when T sub n and S sub n are both lattice random variables. We call our theorems large deviation local limit theorems for R sub n, since the conditions of our theorems imply that R sub n approaches limit of c in probability for some constant c. We present some examples to illustrate our theorems.

Refined Large Deviation Limit Theorems
Refined Large Deviation Limit Theorems

Author: Vladimir Vinogradov

Publisher: CRC Press

Published: 2023-06-14

Total Pages: 228

ISBN-13: 100094834X

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This is a developing area of modern probability theory, which has applications in many areas. This volume is devoted to the systematic study of results on large deviations in situations where Cramér's condition on the finiteness of exponential moments may not be satisfied

Limit Theorems for Large Deviations
Limit Theorems for Large Deviations

Author: L. Saulis

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 241

ISBN-13: 9401135304

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"Et moi ... - si j'avait su comment en revenir. One service mathematics has rendered the je n'y serais poin t aile.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell O.H ea viside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non Iinearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service. topology has rendered mathematical physics .. .':: 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d 'e1:re of this series

Large Deviation Local Limit Theorems for Arbitrary Sequences of Random Variables
Large Deviation Local Limit Theorems for Arbitrary Sequences of Random Variables

Author: Narasinga Rao Chaganty

Publisher:

Published: 1982

Total Pages: 39

ISBN-13:

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The results of W. Richter (Theory Prob. Appl. (1957) 2 206-219) on sums of independent, identically distributed random variables are generalized to arbitrary sequences of random variables Tn. Under simple conditions on the cumulant generating function of Tn, which imply that Tau n/n converges to o, it is shown, for arbitrary sequences (mn) converging to o, that kn(mn), the probability density function of Tn/n at mn, is asymptotic to an expression involving the large deviation rate of Tn/n. Analogous results for lattice random variables are also given. Applications of these results to statistics appearing in nonparametric inference are presented. (Author).

Strong Large Deviation and Local Limit Theorems
Strong Large Deviation and Local Limit Theorems

Author:

Publisher:

Published: 1986

Total Pages: 31

ISBN-13:

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Most large deviation results give asymptotic expressions to log P(Y sub n> or = X sub n) where the event (Y sub n> or = X sub n) is a large deviation event, that is, its probability goes to zero exponentially fast. The authors to such results for arbitrary random variables (Y sub n), that is, it obtains asymptotic expressions for P(Y sub n> or = X sub n) where (Y sub n> or = X sub n) is a large deviation event. These strong large deviation results are obtained for lattice valued and nonlattice valued random variables and require some conditions on their moment generating functions. A result that gives the average probability that Y sub n lies in an interval 2h/b sub n around the point Y sub n where h> 0, b sub n approaches limit of y*, is referred to as a local limit result for (Y sub n). This paper obtains local limit theorems for arbitrary random variables based on easily verifiable conditions on their characteristic functions. These local limit theorems play a major role in the proofs of the strong large deviation results of this paper. These results are illustrated with two typical applications.

Some Limit Theorems in Statistics
Some Limit Theorems in Statistics

Author: R. R. Bahadur

Publisher: SIAM

Published: 1971-01-31

Total Pages: 48

ISBN-13: 0898711754

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A discussion of topics in the theory of large deviations and of aspects of estimation and testing in large samples.

Large Deviations
Large Deviations

Author: S. R. S. Varadhan

Publisher: American Mathematical Soc.

Published: 2016-12-08

Total Pages: 114

ISBN-13: 082184086X

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The theory of large deviations deals with rates at which probabilities of certain events decay as a natural parameter in the problem varies. This book, which is based on a graduate course on large deviations at the Courant Institute, focuses on three concrete sets of examples: (i) diffusions with small noise and the exit problem, (ii) large time behavior of Markov processes and their connection to the Feynman-Kac formula and the related large deviation behavior of the number of distinct sites visited by a random walk, and (iii) interacting particle systems, their scaling limits, and large deviations from their expected limits. For the most part the examples are worked out in detail, and in the process the subject of large deviations is developed. The book will give the reader a flavor of how large deviation theory can help in problems that are not posed directly in terms of large deviations. The reader is assumed to have some familiarity with probability, Markov processes, and interacting particle systems.

Large and Moderate Deviation Limit Theorems for Arbitrary Sequences of Random Variables with Applications
Large and Moderate Deviation Limit Theorems for Arbitrary Sequences of Random Variables with Applications

Author:

Publisher:

Published: 1991

Total Pages: 11

ISBN-13:

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In this research project we have obtained several limit theorems for arbitrary and dependent sequences of random variables. The limit theorems considered in this project fall into three categories namely, Large deviation local limit theorems, Strong large deviation theorems and Strong moderate deviation theorems. These three categories are dependent in the sense they all are subcategories of large deviation theory. The theory of large deviations and its many users are well described in the book by Ellis (1985) and the monograph by Varadhan (1984). This work generalized several classical limit theorems that were obtained in the literature for independent and identically distributed random variables. In the next three sections we outline briefly the technical details of the work done under this contract.

Large Deviation Local Limit Theorems for Random Vectors
Large Deviation Local Limit Theorems for Random Vectors

Author: Narasinga R. Chaganty

Publisher:

Published: 1986

Total Pages: 19

ISBN-13:

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This paper presents a survey of large deviation local limit theorems for random vectors. The authors then establish a more extensive large deviation local limit theorem that requires somewhat weaker conditions even in the special cases proved earlier. (Author).