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The paper provides new upper and lower bounds for the multivariate Laplaceapproximation under weak local assumptions. Their range of validity is alsogiven. An application to an integral arising in the extension of the Dixon'sidentity is presented. The paper both generalizes and complements recentresults by Inglot and Majerski and removes their superfluous assumption onvanishing of the third order partial derivatives of the exponent function.

Simple error bounds for the multivariate Laplace approximation under weak local assumptions