By Karl W. Breitung
This booklet supplies a self-contained creation to the topic of asymptotic approximation for multivariate integrals for either mathematicians and utilized scientists. a suite of result of the Laplace equipment is given. Such equipment are priceless for instance in reliability, information, theoretical physics and data idea. a major exact case is the approximation of multidimensional basic integrals. right here the relation among the differential geometry of the boundary of the mixing area and the asymptotic chance content material is derived. probably the most very important functions of those tools is in structural reliability. Engineers operating during this box will locate the following a whole define of asymptotic approximation equipment for failure likelihood integrals.
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Extra info for Asymptotic Approximations for Probability Integrals
2 Basic Results In this section we consider asymptotic approximations for multivariate Laplace type integrals as/3 --, oc I(r = / h(~) e x p ( ~ 2 f ( x ) ) dx. 1) , J F with F C ~ , h(~) a continuous function and f ( ~ ) a twice continuously differentiable function. Here /3 is a real parameter. We set here as parameter ~2 instead of ,~ deviating from the usual terminology, since in the applications in the following chapters for normal integrals the parameter appears in this form. In the years 1948-1966 the Laplace method was generalized to approximations for multivariate integrals.
E. the asymptotic form of functionals of probability measures, which converge towards Gaussian probability measures, is studied. The asymptotic behavior of I(/3) as /3 --* oc can be studied with similar methods as in the univariate case. But here some additional problems appear and due to this there is no complete theory for the asymptotics of these integrals until now. In this report a summary of results will be given, which should be sufficient for most problems in applications. In principle the asymptotic behavior of these integrals is determined by the structure of the functions f and h and of the integration domain F in the neighborhood of the points or sets where the function f achieves its global maximum with respect to F.
28) ot Using the asymptotic form of f ' we get by integrating that u ~ ~-(x - ~)', x -~ ~. 29) , u~0. 30 / r Therefore x - c~ --. 27) we find then the expansion as u --+ 0 h(x(u))~fTa u ,,, a( _ a),-r. 31) and so we get k(u)- b a 1 Using now Watson's lemma gives the result. 32)  From this result some special cases can be derived. C o r o l l a r y 37 Let f and h be continuous functions on a finite interval [c~,fl]. 33) b) If the global maximum occurs only at ~, h(~) # 0 and f ( x ) is near c~ continuously differentiable with f ' ( c 0 < O, then 1 I(A) ~ h(o~)exp(Af(cr))Alf,(oOi , A ~ oo.
Asymptotic Approximations for Probability Integrals by Karl W. Breitung