Moderate Deviation and Berry-Esseen Bounds in the $p$-Spin Curie-Weiss Model

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.