Asymptotic analysis of random walks: light-tailed distributions
"This is a companion book to Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions by A.A. Borovkov and K.A. Borovkov. Its self-contained systematic exposition provides a highly useful resource for academic researchers and professionals interested in applications of probability in sta...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Weitere Verfasser: | , |
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge ; New York, NY ; Port Melbourne ; New Delhi ; Singapore
Cambridge University Press
2020
|
| Schriftenreihe: | Encyclopedia of mathematics and its applications
176 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Zusammenfassung: | "This is a companion book to Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions by A.A. Borovkov and K.A. Borovkov. Its self-contained systematic exposition provides a highly useful resource for academic researchers and professionals interested in applications of probability in statistics, ruin theory, and queuing theory. The large deviation principle for random walks was first established by the author in 1967, under the restrictive condition that the distribution tails decay faster than exponentially. (A close assertion was proved by S.R.S. Varadhan in 1966, but only in a rather special case.) Since then, the principle has always been treated in the literature only under this condition. Recently, the author jointly with A.A. Mogul'skii removed this restriction, finding a natural metric for which the large deviation principle for random walks holds without any conditions. This new version is presented in the book, as well as a new approach to studying large deviations in boundary crossing problems. Many results presented in the book, obtained by the author himself or jointly with co-authors, are appearing in a monograph for the first time |
| Beschreibung: | Literaturverzeichnis: Seite 410-418 |
| Beschreibung: | xvi, 419 Seiten |
| ISBN: | 9781107074682 |
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| adam_text | Asymptotic Analysis of Random Walks
Light—Tailed Distributions
A A BOROVKOV
Sobolev Institute of Mathematics, Novosibirsk
Translated by
VVU LYA N O V
Lomonosov Moscow State University
and HSE University, Moscow
M V ZHITLUKHIN
Steklov Institute of Mathematics, Moscow
7 CAMBRIDGE
UNIVERSITY PRESS
Introduction
l
Contents
Preliminary results
Deviation function and its properties in the one-
dimensional case
Deviation function and its propenies in the multidimen-
sional case
Chebyshev-type exponential inequalities for sums of
random vectors
Properties of the random variable y = AG) and its
deviation function
The integro-local theorems of Stone and Shepp
and Gnedenko’s local theorem
Approximation of distributions of sums of random variables
The Cramér transform The reduction formula
Limit theorems for sums of random variables in the Cramér
deviation zone The asymptotic density
Supplement to section 2 2
Integro-local theorems on the boundary of the Cramér zone
Integro-local theorems outside the Cramér zone
Supplement to section 2 5 The multidimensional case
The class of distributions 57?,
Large deviation principles
Limit theorems for sums of random variables with
non-homogeneous terms
Asymptotics of the renewal function and related problems
The second deviation function
Sums of non-identically distributed random variables in
the triangular array scheme
page ix
vi
Contents
Boundary crossing problems for random walks
3 1 Limit theorems for the distribution of jumps when the end
of a trajectory is fixed A probabilistic interpretation of the
Cramer transform
3 2 The conditional invariance principle and the law
of the iterated logarithm
3 3 The boundary crossing problem
3 4 The first passage time of a trajectory over a high level and
the magnitude of overshoot
3 5 Asymptotics of the distribution of the first passage time
through a fixed horizontal boundary
3 6 Asymptotically linear boundaries
3 7 Crossing of a curvilinear boundary by a normalised
trajectory of a random walk
3 8 Supplement Boundary crossing problems in the
multidimensional case
3 9 Supplement Analytic methods for boundary crossing
problems with linear boundaries
3 10 Finding the numerical values of large deviation probabilities
Large deviation principles for random walk trajectories
4 1 On large deviation principles in metric spaces
4 2 Deviation functional (or integral) for random walk
trajectories and its properties
4 3 Chebyshev-type exponential inequalities for trajectories of
random walks
4 4 Large deviation principles for continuous random walk
trajectories Strong versions
4 5 An extended problem setup
4 6 Large deviation principles in the space of functions without
discontinuities of the second kind
4 7 Supplement Large deviation principles in the space (V, pv)
4 8 Conditional large deviation principles in the space (11), p)
4 9 Extension of results to processes with independent increments
4 10 On large deviation principles for compound renewal
processes
411 On large deviation principles for sums of random variables
defined on a finite Markov chain
Contents
Moderately large deviation principles for the trajectories of
random walks and processes with independent increments
5 1 Moderately large deviation principles for sums S,l
5 2 Moderately large deviation principles for trajectories 5,,
5 3 Moderately large deviation principles for processes with
independent increments
5 4 Moderately large deviation principle as an extension of the
invariance principle to the large deviation zone
5 5 Conditional moderately large deviation principles for the
trajectories of random walks
Some applications to problems in mathematical statistics
6 1 Tests for two simple hypotheses Parameters
of the most powerful tests
6 2 Sequential analysis
6 3 Asymptotically optimal non-parametric goodness of fit tests
6 4 Appendix On testing two composite parametric hypotheses
6 5 Appendix The change point problem
Basic notation
References
Index
vii
|
| adam_txt |
Asymptotic Analysis of Random Walks
Light—Tailed Distributions
A A BOROVKOV
Sobolev Institute of Mathematics, Novosibirsk
Translated by
VVU LYA N O V
Lomonosov Moscow State University
and HSE University, Moscow
M V ZHITLUKHIN
Steklov Institute of Mathematics, Moscow
7 CAMBRIDGE
UNIVERSITY PRESS
Introduction
l
Contents
Preliminary results
Deviation function and its properties in the one-
dimensional case
Deviation function and its propenies in the multidimen-
sional case
Chebyshev-type exponential inequalities for sums of
random vectors
Properties of the random variable y = AG) and its
deviation function
The integro-local theorems of Stone and Shepp
and Gnedenko’s local theorem
Approximation of distributions of sums of random variables
The Cramér transform The reduction formula
Limit theorems for sums of random variables in the Cramér
deviation zone The asymptotic density
Supplement to section 2 2
Integro-local theorems on the boundary of the Cramér zone
Integro-local theorems outside the Cramér zone
Supplement to section 2 5 The multidimensional case
The class of distributions 57?,
Large deviation principles
Limit theorems for sums of random variables with
non-homogeneous terms
Asymptotics of the renewal function and related problems
The second deviation function
Sums of non-identically distributed random variables in
the triangular array scheme
page ix
vi
Contents
Boundary crossing problems for random walks
3 1 Limit theorems for the distribution of jumps when the end
of a trajectory is fixed A probabilistic interpretation of the
Cramer transform
3 2 The conditional invariance principle and the law
of the iterated logarithm
3 3 The boundary crossing problem
3 4 The first passage time of a trajectory over a high level and
the magnitude of overshoot
3 5 Asymptotics of the distribution of the first passage time
through a fixed horizontal boundary
3 6 Asymptotically linear boundaries
3 7 Crossing of a curvilinear boundary by a normalised
trajectory of a random walk
3 8 Supplement Boundary crossing problems in the
multidimensional case
3 9 Supplement Analytic methods for boundary crossing
problems with linear boundaries
3 10 Finding the numerical values of large deviation probabilities
Large deviation principles for random walk trajectories
4 1 On large deviation principles in metric spaces
4 2 Deviation functional (or integral) for random walk
trajectories and its properties
4 3 Chebyshev-type exponential inequalities for trajectories of
random walks
4 4 Large deviation principles for continuous random walk
trajectories Strong versions
4 5 An extended problem setup
4 6 Large deviation principles in the space of functions without
discontinuities of the second kind
4 7 Supplement Large deviation principles in the space (V, pv)
4 8 Conditional large deviation principles in the space (11), p)
4 9 Extension of results to processes with independent increments
4 10 On large deviation principles for compound renewal
processes
411 On large deviation principles for sums of random variables
defined on a finite Markov chain
Contents
Moderately large deviation principles for the trajectories of
random walks and processes with independent increments
5 1 Moderately large deviation principles for sums S,l
5 2 Moderately large deviation principles for trajectories 5,,
5 3 Moderately large deviation principles for processes with
independent increments
5 4 Moderately large deviation principle as an extension of the
invariance principle to the large deviation zone
5 5 Conditional moderately large deviation principles for the
trajectories of random walks
Some applications to problems in mathematical statistics
6 1 Tests for two simple hypotheses Parameters
of the most powerful tests
6 2 Sequential analysis
6 3 Asymptotically optimal non-parametric goodness of fit tests
6 4 Appendix On testing two composite parametric hypotheses
6 5 Appendix The change point problem
Basic notation
References
Index
vii |
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| spelling | Borovkov, A. A. 1931- Verfasser (DE-588)1089930224 aut Asymptotic analysis of random walks light-tailed distributions A.A. Borovkov, Sobolev Institute of Mathematics, Novosibirs ; translated by V.V. Ulyanov, Lomonosov Moscow State University and HSE University, Moscow, M. V. Zhitlukhin, Steklov Institute of Mathematics, Moscow Cambridge ; New York, NY ; Port Melbourne ; New Delhi ; Singapore Cambridge University Press 2020 xvi, 419 Seiten txt rdacontent n rdamedia nc rdacarrier Encyclopedia of mathematics and its applications 176 Literaturverzeichnis: Seite 410-418 "This is a companion book to Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions by A.A. Borovkov and K.A. Borovkov. Its self-contained systematic exposition provides a highly useful resource for academic researchers and professionals interested in applications of probability in statistics, ruin theory, and queuing theory. The large deviation principle for random walks was first established by the author in 1967, under the restrictive condition that the distribution tails decay faster than exponentially. (A close assertion was proved by S.R.S. Varadhan in 1966, but only in a rather special case.) Since then, the principle has always been treated in the literature only under this condition. Recently, the author jointly with A.A. Mogul'skii removed this restriction, finding a natural metric for which the large deviation principle for random walks holds without any conditions. This new version is presented in the book, as well as a new approach to studying large deviations in boundary crossing problems. Many results presented in the book, obtained by the author himself or jointly with co-authors, are appearing in a monograph for the first time Asymptotik (DE-588)4126634-1 gnd rswk-swf Irrfahrtsproblem (DE-588)4162442-7 gnd rswk-swf Irrfahrtsproblem (DE-588)4162442-7 s Asymptotik (DE-588)4126634-1 s DE-604 Ulʹjanov, Vladimir Vladimirovič 1953- (DE-588)140792015 trl Zhitlukhin, M. V. (DE-588)1221985779 trl 9781139871303 Erscheint auch als Online-Ausgabe Borovkov, A Asymptotic analysis of random walks: light-tailed distributions New York : Cambridge University Press, 2020 Encyclopedia of mathematics and its applications 176 (DE-604)BV000903719 176 V:DE-603;B:DE-17 application/pdf http://scans.hebis.de/HEBCGI/show.pl?47344655_toc.pdf Inhaltsverzeichnis HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032664810&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Borovkov, A. A. 1931- Asymptotic analysis of random walks light-tailed distributions Encyclopedia of mathematics and its applications Asymptotik (DE-588)4126634-1 gnd Irrfahrtsproblem (DE-588)4162442-7 gnd |
| subject_GND | (DE-588)4126634-1 (DE-588)4162442-7 |
| title | Asymptotic analysis of random walks light-tailed distributions |
| title_auth | Asymptotic analysis of random walks light-tailed distributions |
| title_exact_search | Asymptotic analysis of random walks light-tailed distributions |
| title_exact_search_txtP | Asymptotic analysis of random walks light-tailed distributions |
| title_full | Asymptotic analysis of random walks light-tailed distributions A.A. Borovkov, Sobolev Institute of Mathematics, Novosibirs ; translated by V.V. Ulyanov, Lomonosov Moscow State University and HSE University, Moscow, M. V. Zhitlukhin, Steklov Institute of Mathematics, Moscow |
| title_fullStr | Asymptotic analysis of random walks light-tailed distributions A.A. Borovkov, Sobolev Institute of Mathematics, Novosibirs ; translated by V.V. Ulyanov, Lomonosov Moscow State University and HSE University, Moscow, M. V. Zhitlukhin, Steklov Institute of Mathematics, Moscow |
| title_full_unstemmed | Asymptotic analysis of random walks light-tailed distributions A.A. Borovkov, Sobolev Institute of Mathematics, Novosibirs ; translated by V.V. Ulyanov, Lomonosov Moscow State University and HSE University, Moscow, M. V. Zhitlukhin, Steklov Institute of Mathematics, Moscow |
| title_short | Asymptotic analysis of random walks |
| title_sort | asymptotic analysis of random walks light tailed distributions |
| title_sub | light-tailed distributions |
| topic | Asymptotik (DE-588)4126634-1 gnd Irrfahrtsproblem (DE-588)4162442-7 gnd |
| topic_facet | Asymptotik Irrfahrtsproblem |
| url | http://scans.hebis.de/HEBCGI/show.pl?47344655_toc.pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032664810&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000903719 |
| work_keys_str_mv | AT borovkovaa asymptoticanalysisofrandomwalkslighttaileddistributions AT ulʹjanovvladimirvladimirovic asymptoticanalysisofrandomwalkslighttaileddistributions AT zhitlukhinmv asymptoticanalysisofrandomwalkslighttaileddistributions |
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