A note on the smallest eigenvalue of the empirical covariance of causal Gaussian processes

TL;DR

This paper provides a simple proof for bounding the smallest eigenvalue of the empirical covariance in causal Gaussian processes, using elementary Gaussian facts and a causal decomposition, with applications to vector autoregression.

Contribution

It introduces a novel, elementary proof technique for eigenvalue bounds in causal Gaussian processes and applies it to least squares identification of vector autoregression.

Findings

Established a one-sided tail inequality for Gaussian quadratic forms

Provided a performance guarantee for least squares in vector autoregression

Demonstrated the proof's simplicity using elementary Gaussian facts

Abstract

We present a simple proof for bounding the smallest eigenvalue of the empirical covariance in a causal Gaussian process. Along the way, we establish a one-sided tail inequality for Gaussian quadratic forms using a causal decomposition. Our proof only uses elementary facts about the Gaussian distribution and the union bound. We conclude with an example in which we provide a performance guarantee for least squares identification of a vector autoregression.