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The smallest singular value of large random rectangular Toeplitz and circulant matrices

Alexei Onatski, Vladislav Kargin

Abstract

Let $x_i$, $i\in\mathbb{Z}$ be a sequence of i.i.d. standard normal random variables. Consider rectangular Toeplitz $\mathbf{X}=\left(x_{j-i}\right)_{1\leq i\leq p,1\leq j\leq n}$ and circulant $\mathbf{X}=\left(x_{(j-i)\mod n}\right)_{1\leq i\leq p,1\leq j\leq n}$ matrices. Let $p,n\rightarrow\infty$ so that $p/n\rightarrow c\in(0,1]$. We prove that the smallest eigenvalue of $\frac{1}{n}\mathbf{X}\mathbf{X}^\top$ converges to zero in probability and in expectation. We establish a lower bound on the rate of this convergence. The lower bound is faster than any poly-log but slower than any polynomial rate. For the ``rectangular circulant'' matrices, we also establish a polynomial upper bound on the convergence rate, which is a simple explicit function of $c$.

The smallest singular value of large random rectangular Toeplitz and circulant matrices

Abstract

Let , be a sequence of i.i.d. standard normal random variables. Consider rectangular Toeplitz and circulant matrices. Let so that . We prove that the smallest eigenvalue of converges to zero in probability and in expectation. We establish a lower bound on the rate of this convergence. The lower bound is faster than any poly-log but slower than any polynomial rate. For the ``rectangular circulant'' matrices, we also establish a polynomial upper bound on the convergence rate, which is a simple explicit function of .

Paper Structure

This paper contains 9 sections, 8 theorems, 132 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main results and proof outlines
  3. Monte Carlo
  4. Proof details
  5. Proof details for Theorem \ref{['thm: exponential']}
  6. Proof details for Theorem \ref{['thm: expectation']}
  7. Proof details for Theorem \ref{['thm: lower bound']}
  8. Proof details for Theorem \ref{['thm: small T/p']}
  9. Conclusion

Key Result

Theorem 1

Suppose that $p,n\rightarrow_c\infty$, where $c\in(0,1]$. Then, for both Toeplitz and circulant cases, there exists a constant $\beta>0$ that may depend on $c$, such that

Figures (4)

  • Figure 1: Monte Carlo results for random circulant matrix \ref{['circulant matrix']}. The number of MC replications is 10,000. Top panel: $p=100, n=1,000$. Bottom panel: $p=1,000,n=10,000$. Left panel: MC average of all the eigenvalues of $\mathbf{S}$. Right panel: histogram of MC realizations of the smallest eigenvalue.
  • Figure 2: 25,50, and 75 percentiles of the Monte Carlo distribution of $\log \lambda_p$ for $p\times n$ rectangular circulant $\mathbf{X}$. Based on 10,000 MC replications. Equations reported in the figure correspond to the ordinary least squares estimates of the median line based on seven observations (dot markers).
  • Figure 3: 25,50, and 75 percentiles of the Monte Carlo distribution of $\log \lambda_p$ for $p\times n$ rectangular Toeplitz $\mathbf{X}$. Based on 10,000 MC replications. Equations reported in the figure correspond to the ordinary least squares estimates of the median line based on seven observations (dot markers).
  • Figure 4: Illustration of the inequality \ref{['trigonometric variability']}. Adjacent red dots form clusters of $\omega_i$ of the first type. Adjacent black dots form clusters of $\omega_i$ of the second type. Total variation of the smooth graph, representing the left hand of \ref{['trigonometric variability']}, is larger than that of the piece-wise linear dashed graph, representing the right hand side of \ref{['trigonometric variability']}.

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 1: Proposition 6, barnett22
  • Lemma 2
  • Lemma 3
  • Lemma 4