  We build a unifying convex analysis framework characterizing the statistical
properties of a large class of penalized estimators, both under a regular and
an irregular design. Our framework interprets penalized estimators as proximal
estimators, defined by a proximal operator applied to a corresponding initial
estimator. We characterize the asymptotic properties of proximal estimators,
showing that their asymptotic distribution follows a closed-form formula
depending only on (i) the asymptotic distribution of the initial estimator,
(ii) the estimator's limit penalty subgradient and (iii) the inner product
defining the associated proximal operator. In parallel, we characterize the
Oracle features of proximal estimators from the properties of their penalty's
subgradients. We exploit our approach to systematically cover linear regression
settings with a regular or irregular design. For these settings, we build new
$\sqrt{n}-$consistent, asymptotically normal Ridgeless-type proximal
estimators, which feature the Oracle property and are shown to perform
satisfactorily in practically relevant Monte Carlo settings.
