Asymptotic Approximations for Probability Integrals by Karl W. Breitung

By Karl W. Breitung

This publication provides a self-contained advent 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, facts, theoretical physics and data idea. a major designated case is the approximation of multidimensional common integrals. right here the relation among the differential geometry of the boundary of the mixing area and the asymptotic likelihood 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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There exists a neighborhood U of M such that for all x E U always h(x) > o (or h(~) < 0). 5. For all neighborhoods V of M always dx > 0. 3) F and for all neighborhoods V of M holds that f h(~)j's(~) d ~ - f h(~)e~'J(:~) d~, ~,-~ ~. 4) PROOF: Let be m -- supwcF f ( x ) and 5, c be positive constants. 5) e~1(Y) >_ e(X-1)('~-6)e ](y). 6) or In the same way for ~ _> 1 and f ( y ) < m - ~ we find e )~](y) <_ e('~-l)(m-e)e "f(y). 7) in the limit as e ~ 0 that F F This proves the first part of the lemma.

5) hold. 6) m--0 PROOF: Let R be a fixed positive number. Then we write R oo /(A) = / e - ~ t f ( t ) dt+ f e - ~ t f ( t ) dt. 4) follows that there is a constant K such that If(t)] _< Ke at for all t > R. 8) I1=(~)1 _< /e-~'(f(01dt R (DO OO _< / e - ~ t K e a t dt Is / e-(A-a)tdt = - -K e_(~,_~) n A-a = R R FL~ - a e-a"]J = o(e-~R)" The last relation follows, since the term in the square brackets remains bounded as )l---+ oo. 9) m----0 with pn(t) = O(t aN+l) as t ~ 0+. ,. 11) 0 0 _ R F(1 -t- am) + O(e_AR), A ---+c~.

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. Here the boundary of the integration domain is a surface in /R '~.

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