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This paper re-examines the asymptotic performance analysis of second-order methods for parameter estimation in a general context. It provides a unifying framework to investigate the asymptotic performance of second-order methods under the stochastic model assumption in which both the waveforms and noise signals are possibly temporally correlated, possibly non-Gaussian, real, or complex (possibly noncircular) random processes. Thanks to a functional approach and a matrix-valued reformulated central limit theorem about the sample covariance matrix, the conditions under which the asymptotic covariance of a parameter estimator are dependent or independent of the distribution of the signal involved are specified. Finally, we demonstrate the application of our general results to direction of arrival (DOA) estimation, identification of finite impulse response models, sinusoidal frequency estimation for mixed spectra time series, and frequency estimation of sinusoidal signal with very lowpass envelope