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Independent component analysis (ICA) is a popular approach for blind source separation where the mixing process is assumed to be unchanged with a fixed set of stationary source signals. However, the mixing system and source signals are nonstationary in real-world applications, e.g., the source signals may abruptly appear or disappear, the sources may be replaced by new ones or even moving by time. This paper presents an online learning algorithm for the Gaussian process (GP) and establishes a separation procedure in the presence of nonstationary and temporally correlated mixing coefficients and source signals. In this procedure, we capture the evolved statistics from sequential signals according to online Bayesian learning. The activity of nonstationary sources is reflected by an automatic relevance determination, which is incrementally estimated at each frame and continuously propagated to the next frame. We employ the GP to characterize the temporal structures of time-varying mixing coefficients and source signals. A variational Bayesian inference is developed to approximate the true posterior for estimating the nonstationary ICA parameters and for characterizing the activity of latent sources. The differences between this ICA method and the sequential Monte Carlo ICA are illustrated. In the experiments, the proposed algorithm outperforms the other ICA methods for the separation of audio signals in the presence of different nonstationary scenarios.