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Laplace approximation of high dimensional integrals. (English) Zbl 0826.41026

Summary: It is shown that the usual Laplace approximation is not a valid asymptotic approximation when the dimension of the integral is comparable with the limiting parameter \(n\). The formal Laplace expansion for multidimensional integrals is given and used to construct asymptotic approximations for high dimensional integrals. One example is considered in which the dimension of the integral is \(O(n^{1/2})\) and the relative error of the unimodified Laplace approximation is \(O(1)\). Nevertheless, it is possible to construct a valid asymptotic expansion by regrouping terms in the formal expansion according to asymptotic order in \(n\).

MSC:

41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
65C99 Probabilistic methods, stochastic differential equations
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