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This paper studies a linear regression model in which the errors are asymptotically almost negatively associated (AANA, in short) random variables. Firstly, the central limit theorem for AANA sequences of random variables is established. Then, we use the central limit theorem to investigate the asymptotic normality of the M-estimator for the unknown parameters. Some results for independent and negatively associated (NA, in short) random variables are extended to the case of AANA setting. Finally, a simulation is carried out to verify the asymptotic normality of the M-estimator in the model.