Asymptotics For High Dimensional Regression M-Estimates: Fixed Design   Results
  We investigate the asymptotic distributions of coordinates of regression M-estimates in the moderate $p/n$ regime, where the number of covariates $p$ grows proportionally with the sample size $n$. Under appropriate regularity conditions, we establish the coordinate-wise asymptotic normality of regression M-estimates assuming a fixed-design matrix. Our proof is based on the second-order Poincar\'{e} inequality (Chatterjee, 2009) and leave-one-out analysis (El Karoui et al., 2011). Some relevant examples are indicated to show that our regularity conditions are satisfied by a broad class of design matrices. We also show a counterexample, namely the ANOVA-type design, to emphasize that the technical assumptions are not just artifacts of the proof. Finally, the numerical experiments confirm and complement our theoretical results. 
Lihua Lei, Peter J. Bickel, and Noureddine El Karoui
math.ST stat.TH