This paper derives a closed-form expression for the Cramer-Rao bound (CRB) on estimating the source signals in the linear independent component analysis problem, assuming that all independent components have finite variance. It is also shown that the fixed-point algorithm known as FastICA can approach the CRB (the estimate can be nearly efficient) in two situations: (1) when the distribution of the sources is not too much different from Gaussian, for the symmetric version of the algorithm using any of the custom nonlinear functions (pow3, tanh, gauss); (2) when the distribution of the sources is very different from Gaussian (e.g. has long tails) and the nonlinear function in the algorithm equals the score function of each independent component.