Limiting Behavior of M-Estimators of Regression Coefficients in High Dimensional Linear Models I. Scale Dependent Case

Asymptotics of M-estimators of the regression coefficients in linear models (both scale-variant and scale-invariant) when the number of regression coefficients tends to infinity as the sample size increases are investigated The main purpose of this study is to establish the asymptotic properties under weaker conditions than usually assumed, especially to relax the restrictions on the order of the dimension. Also, the conditions assumed and the results obtained seem easy to be extended to the multivariate linear models. In the first part of the paper, the asymptotic behavior of the ordinary (i.e., not scale-invariant) M-estimates is considered.