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.

翻译：我们研究独立同分布随机向量 $X_i\in\mathbb R^d$ 之和在增长维度下的鞍点近似方法。鞍点近似通过矩生成函数为概率密度函数与分布函数提供高精度逼近。Tang 和 Reid 的最新工作将鞍点近似推广到维度 $d$ 随 $n$ 增长的情形，获得了 $O(d^3/n)$ 阶的误差率。我们改进了该分析，将鞍点近似误差率提升至 $O(d^2/n)$。我们得到了乘性鞍点近似误差的非渐近界。作为推论，我们在 $d^2/n \to 0$ 的条件下建立了增长维度中密度函数的首个局部中心极限定理，并给出了显式的乘性误差界。通过高斯混合模型的示例阐明了我们的结果。