# Optimized Tail Bounds for Random Matrix Series

## TL;DR

This paper improves tail bounds for random matrix series by using intrinsic dimension, making them more applicable in high-dimensional settings.

## Contribution

The novelty lies in using intrinsic dimension instead of ambient dimension for tail bounds in random matrix series.

## Key findings

- Modified tail bounds for matrix Gaussian and sub-Gaussian series are derived using intrinsic dimension.
- Expectation bounds for random matrix series are obtained based on intrinsic dimension.
- The approach is suitable for high-dimensional or infinite-dimensional settings.

## Abstract

Random matrix series are a significant component of random matrix theory, offering rich theoretical content and broad application prospects. In this paper, we propose modified versions of tail bounds for random matrix series, including matrix Gaussian (or Rademacher) and sub-Gaussian and infinitely divisible (i.d.) series. Unlike present studies, our results depend on the intrinsic dimension instead of ambient dimension. In some cases, the intrinsic dimension is much smaller than ambient dimension, which makes the modified versions suitable for high-dimensional or infinite-dimensional setting possible. In addition, we obtain the expectation bounds for random matrix series based on the intrinsic dimension.