Windowed cumulant projections of non-Gaussian linear processes yield autocorrelation estimators which are immune to additive Gaussian noise of unknown covariance. By establishing strong consistency of these estimators, strongly consistent and noise insensitive recursive algorithms are developed for parameter estimation. These computationally attractive schemes are shown to be optimal with respect to a modified mean-square-error (MSE) criterion which implicitly exploits the high signal-to-noise ratio domain of cumulant statistics. The novel MSE objective function is expressed in terms of the noisy process, but it is shown to be a scalar multiple of the standard MSE criterion as if the latter was computed in the absence of noise. Simulations illustrate the performance of the proposed algorithms and compare them with the conventional algorithms.