The Laplace asymptotic expansion in high dimensions

Understand

We prove that the classical Laplace asymptotic expansion (AE) of $\int_{\mathbb R^d} g(x)e^{-nu(x)}dx$, $n\gg1$ extends to the high-dimensional regime in which $d$ may grow large with $n$.

• More specifically, we use new techniques suitable to high-$d$ to derive an AE which formally coincides with the classical one because the terms are the same, but which now has a new small parameter.
• Namely under classical assumptions on $z$ and $g$ and additional bounds on the growth of $\|\nabla^kz\|$ and $\|\nabla^kg\|$ with $d$, we show the new small parameter is $d^2/n$, in the sense that $|\text{Rem}_L|\leq C_L(d^2/n)^L$ for each $L=1,2,3,\dots$, where $\text{Rem}_L$ is the $L$th order remainder.
• As an example, we show that the derivative bounds are satisfied with high probability for a random function $z$ arising in a standard statistical model.