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

Somabha Mukherjee, Tianyu Liu, Bhaswar B. Bhattacharya

Abstract

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 Cramér-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.

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

Abstract

Limit theorems for the magnetization in the -spin Curie-Weiss model, for , has been derived recently by Mukherjee et al. (2021). In this paper, we strengthen these results by proving Cramér-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 when the magnetization has asymptotically Gaussian fluctuations, and it is 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 -spin Curie-Weiss model with no external field, for all points in the parameter space where consistent estimation is possible.
Paper Structure (13 sections, 10 theorems, 115 equations)

This paper contains 13 sections, 10 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Organization:
  3. Fluctuations of the Magnetization
  4. Statements of the Main Results
  5. Moderate Deviations for the Magnetization
  6. Berry-Esseen Bounds for the Magnetization
  7. Berry-Esseen Bounds for the Maximum Pseudolikelihood Estimate of $\beta$
  8. Proofs of the Main Results
  9. Proof of Theorem \ref{['thm:moderatedeviationsresult']}
  10. Proof of Theorem \ref{['thm:berryesseen']}
  11. Proof of Theorem \ref{['thm:berryesseenestimate']}
  12. Future Directions
  13. Proof of Lemma \ref{['lembn']}

Key Result

Theorem 1

Fix $p \geqslant 3$ and $(\beta, h) \in \Theta$, and suppose $\bm X \sim \mathbb{P}_{\beta,h,p}$. Then the following hold:

Theorems & Definitions (16)

  • Theorem 1: Theorem 2.1 in smfl
  • Theorem 2
  • Remark 3.1
  • Theorem 3
  • Remark 3.2
  • Theorem 4
  • Theorem 5
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 6 more