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Linear Eigenvalue Statistics of $XX^\prime$ matrices

Kiran Kumar A. S, Shambhu Nath Maurya, Koushik Saha

Abstract

This article focuses on the fluctuations of linear eigenvalue statistics of $T_{n\times p}T'_{n\times p}$, where $T_{n\times p}$ is an $n\times p$ Toeplitz matrix with real, complex or time-dependent entries. We show that as $n \rightarrow \infty$ and $p/n \rightarrow λ\in (0, \infty)$, the linear eigenvalue statistics of these matrices for polynomial test functions converge in distribution to Gaussian random variables. We also discuss the linear eigenvalue statistics of $H_{n\times p}H'_{n\times p}$, when $H_{n\times p}$ is an $n\times p$ Hankel matrix. As a result of our studies, we also derive in-probability limit and a central limit theorem type result for Schettan norm of rectangular Toeplitz matrices. To establish the results, we use method of moments.

Linear Eigenvalue Statistics of $XX^\prime$ matrices

Abstract

This article focuses on the fluctuations of linear eigenvalue statistics of , where is an Toeplitz matrix with real, complex or time-dependent entries. We show that as and , the linear eigenvalue statistics of these matrices for polynomial test functions converge in distribution to Gaussian random variables. We also discuss the linear eigenvalue statistics of , when is an Hankel matrix. As a result of our studies, we also derive in-probability limit and a central limit theorem type result for Schettan norm of rectangular Toeplitz matrices. To establish the results, we use method of moments.
Paper Structure (17 sections, 21 theorems, 134 equations)

This paper contains 17 sections, 21 theorems, 134 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Symmetric Toeplitz with independent entries
  4. Limiting moment sequence
  5. Fluctuation of $T_{n \times p}$ with diagonal entries as zero
  6. Fluctuation of $T_{n \times p}$ with non-zero diagonal entries
  7. Symmetric Toeplitz with Brownian motion entries
  8. Fluctuation of $T_{n \times p}(t)$ with diagonal entries as zero
  9. Finite dimensional convergence:
  10. Tightness:
  11. Fluctuation of $T_{n \times p}(t)$ with non-zero diagonal entries
  12. Non-symmetric Toeplitz matrices
  13. Limiting moment sequence:
  14. Fluctuation for independent entries:
  15. Fluctuation for time-dependent entries:
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\lambda \in (0,\infty)$ and $\{a_i\}_{i \in \mathbb{N}}$ be a sequence of random variables which satisfy Assumption I with $a_0 \equiv 0$. Suppose $T_{n \times p}$ is the $n \times p$ symmetric Toeplitz matrix with input entries $\{\frac{a_i}{\sqrt{n}}\}_{i \geq 0}$. Then for every $k\geq 1$, a where $\{N_{k}; k\geq 1\}$ are zero mean Gaussian distributions with covariance structure as in (eq

Theorems & Definitions (48)

  • Theorem 1
  • Remark 2
  • Theorem 3
  • Remark 4
  • Remark 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Lemma 10
  • ...and 38 more