Rates of Convergence of the Magnetization in the Tensor Curie-Weiss Potts Model

TL;DR

This paper establishes precise convergence rates for the magnetization and inverse temperature estimator in the tensor Curie-Weiss Potts model, revealing different phases with varying rates depending on the parameter space region.

Contribution

It provides Berry-Esseen type bounds for the magnetization vector and the inverse temperature estimator, extending previous limit theorems with explicit convergence rates.

Findings

Rate of convergence is $N^{-1/2}$ in most parameter regions.

At special points, convergence rates are $N^{-1/4}$ or $N^{-1/6}.

Results depend on the behavior of the fourth derivative of the free energy function.

Abstract

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.