We study the classical Laplace asymptotic expansion of $\int_{\mathbb R^d} f(x)e^{-nv(x)}dx$ in high dimensions $d$. We derive an error bound to the expansion when truncated to arbitrary order. The error bound is fully explicit except for absolute constants, and it depends on $d$, $n$, and operator norms of the derivatives of $v$ and $f$ in a neighborhood of the minimizer of $v$.

翻译：我们研究了高维$d$情形下经典拉普拉斯渐近展开$\int_{\mathbb R^d} f(x)e^{-nv(x)}dx$。我们推导了该展开式截断至任意阶时的误差界。该误差界除绝对常数外完全显式，其依赖于$d$、$n$以及$v$的极小值点邻域内$v$和$f$各阶导数的算子范数。