The Laplace asymptotic expansion in high dimensions

Abstract

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. We also show that if the derivative bounds are relaxed, then we still obtain a valid AE in powers of a "larger" small parameter. To prove these results, we derive a very general nonasymptotic bound on $\text{Rem}_L$ which is explicit in its dependence on $g,z,d,n$. The bound holds with nearly no apriori restrictions on the magnitude of the derivative norms. We show the bound is tight for each $L$ by proving a matching lower bound for a quartic $z$ and $g\equiv1$. When $d,z,g$ are fixed and $n\to\infty$, our bound shows that $\text{Rem}_L=O(n^{-L})$. Thus our work subsumes the classical theory of the Laplace expansion, and significantly extends it into the high-$d$ regime. This broadened applicability of the expansion is extremely useful for the many modern applications requiring the computation of high-$d$ Laplace integrals. In settings where the expansion is already in use, our precise and explicit error bound is valuable both for numerical estimates and theoretical analysis, especially near the boundary of applicability of the expansion.