Saddle Point Approximation and Central Limit Theorem for Densities in high dimensions

Abstract: We study the saddlepoint approximation (SPA) for sums of $n$ i.i.d. random vectors $X_i\in\mathbb R^d$ in growing dimensions. SPA provides highly accurate approximations to probability densities and distribution functions via the moment generating function. Recent work by Tang and Reid extended SPA to cases where the dimension $d$ increases with $n$, obtaining an error rate of order $O(d^3/n)$. We refine this analysis and improve the SPA error rate to $O(d^2/n)$. We obtain a non-asymptotic bound for the multiplicative SPA error. As a corollary, we establish the first local central limit theorem for densities in growing dimensions, under the condition $d^2/n \to 0$, and provide explicit multiplicative error bounds. An example involving Gaussian mixtures illustrates our results.