By Karl W. Breitung
This booklet supplies a self-contained advent 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 important for instance in reliability, records, theoretical physics and knowledge concept. a big targeted case is the approximation of multidimensional common integrals. right here the relation among the differential geometry of the boundary of the combination area and the asymptotic chance content material is derived. the most very important purposes of those tools is in structural reliability. Engineers operating during this box will locate the following an entire define of asymptotic approximation tools for failure likelihood integrals.
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Extra info for Asymptotic Approximations for Probability Integrals
55) 0 b(r+h) + f f(x,T + h) dx. 56) 0 + ( b ( ~ + h) - b ( r ) ) f ( x + ~*(b(r + h) - b(~)), ~ + h). Here 0 _< v~(x) _< 1 and 0 _< ~* _< 1. This gives then for the difference quotient I ( r + h) - I(r) h / = J f ( x , ~ + h ~ ( x ) ) dx 0 b(T + h) h w b ( r ) f ( x + O*(b(T + h) - b(v)), v + h). 58) f(x, r) dx, h ---* O, 0 and for the second term we have as h --~ 0 that b(r+h)- h b(r) f ( x + O * ( b ( r + h ) - b(r)),r+ h ) ~ b ' ( r ) f ( r ) . 59) This gives the result in this case. D This result is generalized to functions of several variables in the following theorem.
N)T w i t h tl = [c:[-1c and then a new random vector Y is defined by Y = T X . 105) where the term in the brackets is the first absolute moment of a normally distributed random variable Y1 with mean zero and variance [c[-2cTA-lc; so we get for the integral = lc142var(Yx) = 4 2 1 c T A-1cl. 106) the result is obtained. D C o n v e r g e n c e of P r o b a b i l i t y D i s t r i b u t i o n s D e f i n i t i o n 10 A sequence F,~ of k-dimensional distribution functions is called weakly convergent to a distribution function F iff limoo F .
Here the asymptotic behavior of the functions I(~) or J(A) is studied as A approaches infinity or zero. In the following we will consider only Laplace type integrals. For these integrals the asymptotic behavior is dominated by the structure of the functions near the global maximum points of f with respect to F. In the following we will consider only Laplace type integrals. In the case of Fourier integrals also other stationary points of this functions may be important for the asymptotic behavior of the integrals, therefore the structure of the functions near all points with ~Tf(a) = o has to be considered.