  Linear Least Squares is a very well known technique for parameter estimation,
which is used even when sub-optimal, because of its very low computational
requirements and the fact that exact knowledge of the noise statistics is not
required. Surprisingly, bounding the probability of large errors with finitely
many samples has been left open, especially when dealing with correlated noise
with unknown covariance. In this paper we analyze the finite sample performance
of the linear least squares estimator. Using these bounds we obtain accurate
bounds on the tail of the estimator's distribution. We show the fast
exponential convergence of the number of samples required to ensure a given
accuracy with high probability. We analyze a sub-Gaussian setting with a fixed
or random design matrix of the linear least squares problem. We also extend the
results to the case of a martingale difference noise sequence. Our analysis
method is simple and uses simple $L_{\infty}$ type bounds on the estimation
error. We also provide probabilistic finite sample bounds on the estimation
error $L_2$ norm. The tightness of the bounds is tested through simulation. We
demonstrate that our results are tighter than previously proposed bounds for
$L_{\infty}$ norm of the error. The proposed bounds make it possible to predict
the number of samples required for least squares estimation even when the least
squares is sub-optimal and is used for computational simplicity.
