In this paper, we derive distributional convergence rates for the magnetization vector and the maximum pseudolikelihood estimator of the inverse temperature parameter in the tensor Curie-Weiss Potts model. Limit theorems for the magnetization vector have been derived recently in Bhowal and Mukherjee (2023), where several phase transition phenomena in terms of the scaling of the (centered) magnetization and its asymptotic distribution were established, depending upon the position of the true parameters in the parameter space. In the current work, we establish Berry-Esseen type results for the magnetization vector, specifying its rate of convergence at these different phases. At most points in the parameter space, this rate is $N^{-1/2}$ ($N$ being the size of the Curie-Weiss network), while at some "special" points, the rate is either $N^{-1/4}$ or $N^{-1/6}$, depending upon the behavior of the fourth derivative of a certain "negative free energy function" at these special points. These results are then used to derive Berry-Esseen type bounds for the maximum pseudolikelihood estimator of the inverse temperature parameter whenever it lies above a certain criticality threshold.

翻译：本文推导了张量Curie-Weiss Potts模型中磁化强度向量以及逆温度参数的最大伪似然估计量的分布收敛速率。磁化强度向量的极限定理最近由Bhowal和Mukherjee（2023）建立，他们根据真实参数在参数空间中的位置，确立了（中心化）磁化强度的多种标度行为及其渐近分布所对应的相变现象。在当前工作中，我们针对磁化强度向量建立了Berry-Esseen型结果，明确了其在这些不同相变点处的收敛速率。在参数空间的大多数点处，该收敛速率为$N^{-1/2}$（$N$为Curie-Weiss网络的规模）；而在某些"特殊"点处，收敛速率则根据特定"负自由能函数"在这些特殊点处的四阶导数行为，分别为$N^{-1/4}$或$N^{-1/6}$。这些结果随后被用于推导逆温度参数的最大伪似然估计量在超过特定临界阈值时的Berry-Esseen型界。