A note on the smallest eigenvalue of the empirical covariance of causal Gaussian processes
This work addresses a theoretical issue in statistical learning for Gaussian processes, but it appears incremental as it offers a simplified proof rather than a new result.
The paper tackles the problem of bounding the smallest eigenvalue of the empirical covariance in causal Gaussian processes, providing a simple proof using elementary Gaussian facts and a causal decomposition, with an example application to least squares identification of vector autoregression.
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.