Saddle Point Approximation and Central Limit Theorem for Densities in high dimensions
Alexander Katsevich
TL;DR
This paper advances multivariate saddlepoint approximation in the regime where dimension $d$ grows with sample size $n$ by establishing a non-asymptotic $O(d^2/n)$ SPA error bound and a local central limit theorem for densities under $d^2/n\to0$. The analysis blends inverse Laplace transform techniques with concentration-type arguments to control both tail and local contributions for a complex-valued exponent, and it provides explicit multiplicative error bounds. A general SPA result for arbitrary sequences $Y_n$ (Theorem mainII) extends the framework beyond i.i.d. sums, with a careful discussion of domain analyticity. An illustrative symmetric Gaussian mixture example demonstrates the gains over prior $O(d^3/n)$ results and confirms the local CLT in growing dimensions, with explicit bounds on derivatives governing the error terms.
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.