Limit theorems for the magnetization in the $p$-spin Curie-Weiss model, for $p \geq 3$, has been derived recently by Mukherjee et al. (2021). In this paper, we strengthen these results by proving Cram\'er-type moderate deviation theorems and Berry-Esseen bounds for the magnetization (suitably centered and scaled). In particular, we show that the rate of convergence is $O(N^{-\frac{1}{2}})$ when the magnetization has asymptotically Gaussian fluctuations, and it is $O(N^{-\frac{1}{4}})$ when the fluctuations are non-Gaussian. As an application, we derive a Berry-Esseen bound for the maximum pseudolikelihood estimate of the inverse temperature in $p$-spin Curie-Weiss model with no external field, for all points in the parameter space where consistent estimation is possible.

翻译：Mukherjee等人(2021)近期推导了$p \geq 3$的$p$-旋转Curie-Weiss模型中磁化强度的极限定理。本文通过证明磁化强度(经适当中心化和缩放后)的Cramér型中度偏差定理和Berry-Esseen界，强化了这些结果。特别地，我们证明当磁化强度具有渐近高斯波动时收敛速度为$O(N^{-\frac{1}{2}})$，而当波动为非高斯型时收敛速度为$O(N^{-\frac{1}{4}})$。作为应用，我们推导了无外场$p$-旋转Curie-Weiss模型中逆温度最大伪似然估计的Berry-Esseen界，该结果在参数空间中所有可实现一致估计的点处均成立。