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We present a simple proof for bounding the smallest eigenvalue of theempirical covariance in a causal Gaussian process. Along the way, we establisha one-sided tail inequality for Gaussian quadratic forms using a causaldecomposition. Our proof only uses elementary facts about the Gaussiandistribution and the union bound. We conclude with an example in which weprovide a performance guarantee for least squares identification of a vectorautoregression.

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