Probability

Asymptotic Approximations for Probability Integrals by Karl W. Breitung

By Karl W. Breitung

This e-book offers a self-contained creation to the topic of asymptotic approximation for multivariate integrals for either mathematicians and utilized scientists. a set of result of the Laplace equipment is given. Such tools are beneficial for instance in reliability, records, theoretical physics and knowledge idea. an incredible exact case is the approximation of multidimensional basic integrals. the following the relation among the differential geometry of the boundary of the mixing area and the asymptotic chance content material is derived. the most vital functions of those tools is in structural reliability. Engineers operating during this box will locate the following a whole define of asymptotic approximation tools for failure likelihood integrals.

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R ) dx. g(X,r) 0 in a neighborhood G~(r) = {~; minyeG(r ) lY--~[ < c} of G(r) a coordinate system can be introduced where each y E G,(7") which is given in the form y = ~ + 5 "nr (~) (here nr (~) denotes the surface normal at ~) with x E G(r) has the coordinates (~, 5). In this coordinate system the difference D2(h) can be written in the form D2(h) l(fh) ] f ( ~ + 5 n r ( X ) , v ) D ( x , 5 ) d6 dsr(~). 64) Here D(~, 5) is the transformation determinant for the change of the coordinates.

111) For an arbitrary normal distribution N(p, (r~) instead we have M(s) = e ('a)2/~+s" . 113) The convergence of random variables can be shown by proving the convergence of the corresponding moment generating functions. e. the random variables X,~ converge in distribution towards X . PROOF: See [13], p. 345. [] By combining the last two theorems a convergence theorems for multivariate random vectors can be derived. 32 T h e o r e m 29 A sequence (Xn)neZ~r of k-dimensional random vectors X,~ -~ ( X ~ I , .

G(X,r) 0 in a neighborhood G~(r) = {~; minyeG(r ) lY--~[ < c} of G(r) a coordinate system can be introduced where each y E G,(7") which is given in the form y = ~ + 5 "nr (~) (here nr (~) denotes the surface normal at ~) with x E G(r) has the coordinates (~, 5). In this coordinate system the difference D2(h) can be written in the form D2(h) l(fh) ] f ( ~ + 5 n r ( X ) , v ) D ( x , 5 ) d6 dsr(~). 64) Here D(~, 5) is the transformation determinant for the change of the coordinates.

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