Download Asymptotic Approximations for Probability Integrals by Karl W. Breitung PDF

By Karl W. Breitung

This booklet offers 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 tools is given. Such tools are invaluable for instance in reliability, information, theoretical physics and knowledge thought. a big distinctive case is the approximation of multidimensional general integrals. the following the relation among the differential geometry of the boundary of the mixing area and the asymptotic chance content material is derived. essentially the most very important functions of those equipment is in structural reliability. Engineers operating during this box will locate right here a whole define of asymptotic approximation tools for failure likelihood integrals.

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IX and [127], chap. 8. Additional to the standard results in this report we also treat the case that the maximum of the function is a submanifold of the boundary of the integration domain and that there is an additional function depending on the parameter/3. To avoid unnecessary technical details, we first prove a lemma that shows that under slight regularity conditions it is often possible to restrict the integration domain to a compact set. The following lemma gives conditions under which it is possible to replace an integral over a non-compact domain by an integral over a compact domain without changing its asymptotic behavior.

45 The arn'S are not necessarily integers. 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+.

Therefore before making an expansion we have to choose a suitable scale. This is done usually by making some rough prior estimates of the order of magnitude of the function under consideration. There are normally three (or two) steps in deriving an asymptotic expansion for a function f ( z ) as x ---* x0: 1. Choose an asymptotic scale {On(X)}, 2. 47) n----1 3. 48) n----1 for N = 1 , 2 , . . Often there is no possibility to find the error estimates in step 3. Then judgements about the quality of the approximations can be based only on numerical examples, which give some idea about the magnitude of the error.

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