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Author: Narasinga R. Chaganty
Publisher:
Published: 1986
Total Pages: 19
ISBN-13:
DOWNLOAD EBOOKThis 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).
Author: Narasinga Rao Chaganty
Publisher:
Published: 1982
Total Pages: 39
ISBN-13:
DOWNLOAD EBOOKThe 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).
Author: L. Saulis
Publisher: Springer Science & Business Media
Published: 2012-12-06
Total Pages: 241
ISBN-13: 9401135304
DOWNLOAD EBOOK"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
Author: Vladimir Vinogradov
Publisher: CRC Press
Published: 2023-06-14
Total Pages: 228
ISBN-13: 100094834X
DOWNLOAD EBOOKThis 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
Author: Narasinga Rao Chaganty
Publisher:
Published: 1987
Total Pages: 204
ISBN-13:
DOWNLOAD EBOOKThis 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.
Author:
Publisher:
Published: 1992
Total Pages: 26
ISBN-13:
DOWNLOAD EBOOKWe obtain a strong large deviation result for arbitrary sequence of random vectors under simple and verifiable conditions on the moment generating functions. The key to this result is a local limit theorem for arbitrary sequences of random vectors which is also provided in this paper. The local limit theorem gives conditions on the characteristic functions of random vectors for their pseudo-density function to converge uniformly on bounded sets. We apply these results to the multivariate F-distribution. Large Deviations, Limit Theorems.
Author:
Publisher:
Published: 1986
Total Pages: 31
ISBN-13:
DOWNLOAD EBOOKMost 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.
Author: A.D. Wentzell
Publisher: Springer Science & Business Media
Published: 2012-12-06
Total Pages: 192
ISBN-13: 9400918526
DOWNLOAD EBOOKIn recent decades a new branch of probability theory has been developing intensively, namely, limit theorems for stochastic processes. As compared to classical limit theorems for sums of independent random variables, the generalizations are going here in two directions simultaneously. First, instead of sums of independent variables one considers stochastic processes belonging to certain broad classes. Secondly, instead of the distribution of a single sum - the distribution of the value of a stochastic process at one (time) point - or the joint distribution of the values of a process at a finite number of points, one considers distributions in an infinite-dimensional function space. For stochastic processes constructed, starting from sums of independent random variables, this is the same as considering the joint distribution of an unboundedly increasing number of sums.
Author:
Publisher:
Published: 1991
Total Pages: 11
ISBN-13:
DOWNLOAD EBOOKIn 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.
Author: Alexander Bulinski
Publisher: World Scientific
Published: 2007-09-05
Total Pages: 447
ISBN-13: 9814474576
DOWNLOAD EBOOKThis volume is devoted to the study of asymptotic properties of wide classes of stochastic systems arising in mathematical statistics, percolation theory, statistical physics and reliability theory. Attention is paid not only to positive and negative associations introduced in the pioneering papers by Harris, Lehmann, Esary, Proschan, Walkup, Fortuin, Kasteleyn and Ginibre, but also to new and more general dependence conditions. Naturally, this scope comprises families of independent real-valued random variables. A variety of important results and examples of Markov processes, random measures, stable distributions, Ising ferromagnets, interacting particle systems, stochastic differential equations, random graphs and other models are provided. For such random systems, it is worthwhile to establish principal limit theorems of the modern probability theory (central limit theorem for random fields, weak and strong invariance principles, functional law of the iterated logarithm etc.) and discuss their applications.There are 434 items in the bibliography.The book is self-contained, provides detailed proofs, for reader's convenience some auxiliary results are included in the Appendix (e.g. the classical Hoeffding lemma, basic electric current theory etc.).
