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Linear Least Squares is a very well known technique for parameter estimation,which is used even when sub-optimal, because of its very low computationalrequirements and the fact that exact knowledge of the noise statistics is notrequired. Surprisingly, bounding the probability of large errors with finitelymany samples has been left open, especially when dealing with correlated noisewith unknown covariance. In this paper we analyze the finite sample performanceof the linear least squares estimator under sub-Gaussian martingale differencenoise. In order to analyze this important question we used concentration ofmeasure bounds. When applying these bounds we obtained tight bounds on the tailof the estimator's distribution. We show the fast exponential convergence ofthe number of samples required to ensure a given accuracy with highprobability. We provide probability tail bounds on the estimation error's norm.Our analysis method is simple and uses simple $L_{\infty}$ type bounds on theestimation error. The tightness of the bounds is tested through simulation. Theproposed bounds make it possible to predict the number of samples required forleast squares estimation even when least squares is sub-optimal and used forcomputational simplicity. The finite sample analysis of least squares modelswith this general noise model is novel.

Finite sample performance of linear least squares estimators under sub-Gaussian martingale difference noise