Asymptotic properties for M-estimators in linear models with dependent random errors

• We consider M-estimators of the regression parameter in a linear model when the error process has a specific Markov dependence structure.
• The score function defining the M-estimators is allowed to be non-smooth.
• We establish the moderate deviations, strong Bahadur representations and law of the iterated logarithm for the estimators.
• Our proofs are based on large deviation techniques for special martingale difference arrays.

In this paper, we make use of the technique of martingales to establish the moderate deviations and strong Bahadur representations for M-estimators of the regression parameter in a linear model when the errors form a type of Markov chain. As an application, we obtain a law of the iterated logarithm for the estimators.