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Fluctuation of linear eigenvalue statistics of reverse circulant matrices with independent entries

Shambhu Nath Maurya, Koushik Saha

Abstract

In this article, we study the fluctuations of linear eigenvalue statistics of reverse circulant $(RC_n)$ matrices with independent entries which satisfy some moment conditions. We show that $\frac{1}{\sqrt{n}} \text{Tr} φ(RC_n)$ obey the central limit theorem (CLT) type result, where $φ$ is a nice test function.

Fluctuation of linear eigenvalue statistics of reverse circulant matrices with independent entries

Abstract

In this article, we study the fluctuations of linear eigenvalue statistics of reverse circulant matrices with independent entries which satisfy some moment conditions. We show that obey the central limit theorem (CLT) type result, where is a nice test function.

Paper Structure

This paper contains 3 sections, 4 theorems, 102 equations, 3 figures.

Table of Contents

  1. introduction and main results
  2. Proof of Theorem \ref{['thm:revcircovar']}
  3. Proof of Theorem \ref{['thm:revcirpoly']}

Key Result

Theorem \oldthetheorem

Suppose $RC_n$ is the reverse circulant matrix with independent input sequence $\{\frac{x_i}{\sqrt n}\}_{i\geq 1}$ such that Then for $p,q \geq1$, where Moreover, if $p=q$ then we denote $\sigma_{p,q}$ by $\sigma^{2}_{p}$.

Figures (3)

  • Figure 1: Connection between $J_1,J_2$ and $J_3$.
  • Figure 2: Connection between $J_1,J_2$ and $J_3$: Case I.
  • Figure 3: Connection between $J_1,J_2, \ldots J_k, J_{k+1}$

Theorems & Definitions (17)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • proof : Proof of Theorem \ref{['thm:revcircovar']}
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • ...and 7 more