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The least-mean-square (LMS) is promising in a system whose signal statistics are time-varying, and it is preferred in adaptive digital predistortion techniques. In a nonlinear memory system, it is found that the high nonlinearity, the finite sampling rate and the finite number of digits will introduce colored noise that is dependent on signals. In this paper, the long-term stability of the conventional LMS in the presence of signal-dependent noise is analyzed using perturbation analysis. It is revealed that the cumulative effect of the signal-dependent noise may lead to a divergence of the conventional LMS, and a properly selected leaky coefficient is needed to suppress the divergence. Unfortunately, the compensation performance may be harmed by enlarging the leaky coefficient. To address this problem, a modified LMS (MLMS) algorithm is proposed. By adjusting the updating strategy in the iterations, the long-term stability of the algorithm is guaranteed, without harming the linearization performance. Numerical simulations verify that the MLMS outperforms the conventional LMS both in stability and compensation performance.