# A Uniform Bound on the Operator Norm of Sub-Gaussian Random Matrices and Its Applications

## TL;DR

This paper establishes a uniform bound on the operator norm of sub-Gaussian random matrices with dependent entries, useful for statistical estimation and factor analysis in high-dimensional settings.

## Contribution

It provides a novel uniform bound involving Talagrand's functional for matrices with dependent sub-Gaussian entries, extending previous results to more complex data structures.

## Key findings

- Bound applies to matrices with weakly dependent sub-Gaussian entries.
- The bound incorporates the complexity of the parameter space via Talagrand's functional.
- Applications include operator norm minimization in moment condition estimation and functional data factor analysis.

## Abstract

For an $N \times T$ random matrix $X(\beta)$ with weakly dependent uniformly sub-Gaussian entries $x_{it}(\beta)$ that may depend on a possibly infinite-dimensional parameter $\beta\in \mathbf{B}$, we obtain a uniform bound on its operator norm of the form $\mathbb{E} \sup_{\beta \in \mathbf{B}} ||X(\beta)|| \leq CK \left(\sqrt{\max(N,T)} + \gamma_2(\mathbf{B},d_\mathbf{B})\right)$, where $C$ is an absolute constant, $K$ controls the tail behavior of (the increments of) $x_{it}(\cdot)$, and $\gamma_2(\mathbf{B},d_\mathbf{B})$ is Talagrand's functional, a measure of multi-scale complexity of the metric space $(\mathbf{B},d_\mathbf{B})$. We illustrate how this result may be used for estimation that seeks to minimize the operator norm of moment conditions as well as for estimation of the maximal number of factors with functional data.