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

Ingvar Ziemann

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arXiv · 2212.09508

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

Ingvar Ziemann

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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.

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Ingvar Ziemann. 2022-12-19. A note on the smallest eigenvalue of the empirical covariance of causal Gaussian processes. https://arxiv.org/abs/2212.09508

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