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We study the behavior of high-dimensional robust regression estimators in theasymptotic regime where $p/n$ tends to a finite non-zero limit. Morespecifically, we study ridge-regularized estimators, i.e$\widehat{\beta}=\text{argmin}_{\beta \in \mathbb{R}^p} \frac{1}{n}\sum_{i=1}^n\rho(\varepsilon_i-X_i' \beta)+\frac{\tau}{2}\lVert\beta\rVert^2$. In arecently published paper, we had developed with collaborators probabilisticheuristics to understand the asymptotic behavior of $\widehat{\beta}$. We givehere a rigorous proof, properly justifying all the arguments we had given inthat paper. Our proof is based on the probabilistic heuristics we haddeveloped, and hence ideas from random matrix theory, measure concentration andconvex analysis. While most the work is done for $\tau>0$, we show that undersome extra assumptions on $\rho$, it is possible to recover the case $\tau=0$as a limiting case. We require that the $X_i$'s be i.i.d with independententries, but our proof handles the case where these entries are not Gaussian. A2-week old paper of Donoho and Montanari [arXiv:1310.7320] studied a similarproblem by a different method and with a different point of view. At thispoint, their interesting approach requires Gaussianity of the design matrix.

Asymptotic behavior of unregularized and ridge-regularized high-dimensional robust regression estimators : rigorous results