= Localization condition for Laplace's method
A unique nondegenerate interior minimum does not by itself ensure that a integral treated by <Laplace's method> is localized there. A sufficient condition is a positive gap above the minimum outside every sufficiently small fixed neighbourhood, together with finiteness of the integral for some positive exponent. The exterior contribution is then exponentially negligible. Without localization, $g(y)=y^2(1-y)^4$ on $(-1,1)$ has its unique interior minimum at zero with $g''(0)=2$, but its boundary contribution is at least $e^{-1}n^{-1/4}$ and dominates the nominal $n^{-1/2}$ interior Gaussian contribution.
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