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Asymptotic Approximations for Probability Integrals : Lecture Notes in Mathematics - Karl W. Breitung

Asymptotic Approximations for Probability Integrals

Lecture Notes in Mathematics

Paperback Published: 16th December 1994
ISBN: 9783540586173
Number Of Pages: 154

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This book gives a self-contained introduction to the subject of asymptotic approximation for multivariate integrals for both mathematicians and applied scientists. A collection of results of the Laplace methods is given. Such methods are useful for example in reliability, statistics, theoretical physics and information theory. An important special case is the approximation of multidimensional normal integrals. Here the relation between the differential geometry of the boundary of the integration domain and the asymptotic probability content is derived. One of the most important applications of these methods is in structural reliability. Engineers working in this field will find here a complete outline of asymptotic approximation methods for failure probability integrals.

Introductionp. 1
The Evaluation of Multivariate Integralsp. 1
Structural Reliabilityp. 2
Stochastic Optimizationp. 6
Large Deviations and Extreme Valuesp. 7
Mathematical Statisticsp. 7
Mathematical Preliminariesp. 9
Results from Linear Algebrap. 9
Results from Analysisp. 11
Manifoldsp. 14
Extrema under Constraintsp. 16
Parameter Dependent Integralsp. 21
Probability Distributionsp. 28
Convergence of Probability Distributionsp. 31
Asymptotic Analysisp. 34
The Topic of Asymptotic Analysisp. 34
The Comparison of Functionsp. 35
Asymptotic Power Series and Scalesp. 40
Deriving Asymptotic Expansionsp. 44
Univariate Integralsp. 45
Watson's Lemmap. 45
The Laplace Method for Univariate Functionsp. 47
Multivariate Laplace Type Integralsp. 51
Basic Resultsp. 52
Interior Maximap. 55
Boundary Maximap. 65
Approximations for Normal Integralsp. 85
Time-Invariant Reliability Problemsp. 85
Linear Approximations (FORM Concepts)p. 86
SORM Conceptsp. 90
Asymptotic SORM for Normal Integralsp. 91
The Asymptotic Distribution in the Failure Domainp. 99
Numerical Procedures and Improvementsp. 102
Examplesp. 103
Arbitrary Probability Integralsp. 106
Problems of the Transformation Methodp. 106
Asymptotic Approximations in the Original Spacep. 109
Sensitivity analysisp. 115
Crossing Rates of Stochastic Processesp. 121
Definition and Properties of Stochastic Processesp. 121
Maxima and Crossing of a Stochastic Processp. 124
Crossings through an Hypersurfacep. 127
A Bibliographyp. 135
B Indexp. 145
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783540586173
ISBN-10: 3540586172
Series: Lecture Notes in Mathematics
Audience: General
Format: Paperback
Language: English
Number Of Pages: 154
Published: 16th December 1994
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.39 x 15.6  x 0.89
Weight (kg): 0.24

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