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

TL;DR

This paper refines saddlepoint approximation error bounds for high-dimensional sums of i.i.d. vectors, establishing a local CLT with explicit error rates as dimension grows with sample size.

Contribution

It improves the error rate of saddlepoint approximation in high dimensions from O(d^3/n) to O(d^2/n) and establishes the first local CLT with explicit bounds under growing dimension.

Findings

Refined SPA error rate to O(d^2/n)

Established a non-asymptotic bound for SPA error

Proved a local CLT 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.