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Robust diffusion adaptive estimation algorithms based on the maximum correntropy criterion (MCC), including adapt then combine MCC and combine then adapt MCC, are developed to deal with the distributed estimation over network in impulsive (long-tailed) noise environments. The cost functions used in distributed estimation are in general based on the mean square error (MSE) criterion, which is desirable when the measurement noise is Gaussian. In non-Gaussian situations, especially for the impulsive-noise case, MCC based methods may achieve much better performance than the MSE methods as they take into account higher order statistics of error distribution. The proposed methods can also outperform the robust diffusion least mean p-power (DLMP) and diffusion minimum error entropy (DMEE) algorithms. The mean and mean square convergence analysis of the new algorithms are also carried out. [ABSTRACT FROM AUTHOR]

• The steady-state complementary mean square deviation of the WL-RLS algorithm is analyzed. • The Lyapunov stability analysis of the WL-RLS algorithm for time-invariant system and noise-free output is performed. • Numerical stability analysis has been conducted, illustrating that the WL-RLS algorithm is numerically stable for λ < 1. The widely linear recursive least squares (WL-RLS) algorithm is useful for dealing with noncircular signals. In this paper, we focus on the theoretical prediction and stability of the WL-RLS algorithm. First, the steady-state complementary mean square deviation (CMSD) behavior under a time-varying system is analyzed, which is different from the traditional standard mean square analysis that takes the form of real-valued evolution. Second, the Lyapunov stability and numerical stability are taken into account to provide an in-depth understanding of the stability issue. Specifically, by employing the Lyapunov stability theory (LST) approach, we perform the stability analysis of the WL-RLS algorithm for time-invariant system and noise-free output, illustrating that the WL-RLS algorithm is exponentially convergent in the sense of Lyapunov. Moreover, the numerical stability is analyzed by considering the finite word-length effects, and it is shown that the WL-RLS algorithm is numerically stable when the forgetting factor λ < 1. Numerical simulations for system identification scenarios verify the accuracy of the steady-state theoretical prediction. [ABSTRACT FROM AUTHOR]