Asymptotic behavior for S–estimators in random design linear model with long–range dependent errors ∗ Lin Zhengyan, Li Degui, ∗∗ Chen Jia Department of Mathematics, Zhejiang University, Hangzhou 310027, China Abstract The asymptotic behavior of S–estimators in a random design linear model with long–range dependent Gaussian errors is considered. It turns out that the S– estimators of regression parameter and error variance are strongly consistent under mild conditions. Furthermore, the asymptotic distribution of the S–estimator of regression parameter is normal if the design vectors are i.i.d. and is non–normal if the design vectors are long–range dependent Gaussian vectors. We also show that the asymptotic distribution of S–estimator of the error variance is non–normal since the errors are long–range dependent. Keywords Asymptotic distribution, consistency, linear model, long–range dependence, S–estimator. 2000 MR Subject Classification 62M10, 60F05. 1 Introduction Consider the linear regression model Y i = X T i β 0 + ε i , i =1, 2, ··· , n, (1.1) where {Y i } is a sequence of dependent observations, {X i } is a sequence of d–dimensional random vectors, β 0 is a d–dimensional regression parameter and {ε i } is a stationary * Supported by National Natural Science Foundation of China (Grant No. 10571159) and Specialized Research Fund for the Doctor Program of Higher Education (Grant No. 2002335090) ** Corresponding author: ldgofzju@tom.com. 1