Asymptotic Approximations for Probability Integrals by Karl W. Breitung

By Karl W. Breitung

This e-book 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 tools is given. Such tools are beneficial for instance in reliability, statistics, theoretical physics and data thought. a big certain case is the approximation of multidimensional general integrals. the following the relation among the differential geometry of the boundary of the combination area and the asymptotic chance content material is derived. probably the most vital purposes of those equipment is in structural reliability. Engineers operating during this box will locate the following an entire define of asymptotic approximation equipment for failure likelihood integrals.

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O((X 0~)r - l ) with r > O. -a(x - - - 2. 18) with h(x) = b ( x - a)s-1 + o ( ( x - ~)s-1) with s > O. 19) 3. 20) Gt is finite. 21) o, with A > 1 are all finite and have the asymptotic approximation 8 I(~) ,~ b F r r r A--~ exp(~f(c~)), a ---+ oo. 22) PP~OOF: For simplicity we assume that f ( a ) = 0. To show that all integrals I(A) are finite, we get an upper bound for them II(~)1 ~ /Ih(x){exp(~f(x)) dx t~ P <_ f [h(x)lexp(Af(x)) dx + f Ih(x)lexp(Af(x)) dx. 23) The first integral is finite. 24) o'd-e = Ih(x)l exp (f(x)) dx = O(exp(-)~6)).

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 , . . , Xnk) converges in distribution to a random vector X = ( X 1 , . . , Xk) iff for all t = ( t l , .

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. 34) c) If the global maximum occurs only at ~, h(o0 # 0 and f ( x ) is twice continuously differentiable near ct with f'(c~) = 0 and f " ( ~ ) < O, then Z(,~) ~ h(c~)exp(,~f(c~)) PROOF: 2,Xlf,,(cO [ , ,~ ~ ~ . 35) T h e results follow directly from the last theorem.

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Asymptotic Approximations for Probability Integrals by Karl W. Breitung
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