  We consider an estimator $\hbbeta_n(t)$ defined as the element
$\bphi\in\bPhi$ minimizing a contrast process $\pencontrast(\bphi, t)$ for each
t. We give some general results for deriving the weak convergence of
$\sqrt{n}(\hbbeta_n-\bbeta)$ in the space of bounded functions, where, for each
t, $\bbeta(t)$ is the $\bphi\in\bPhi$ minimizing the limit of
$\pencontrast(\bphi, t)$ as $n\to\infty$. These results are applied in the
context of penalized M-estimation, that is, when $\pencontrast(\bphi,
t)=M_n(\bphi)+ t J_n(\bphi)$, where $M_n$ is a usual contrast process and $J_n$
a penalty such as the $\ell^1$ norm or the squared $\ell^2$ norm. The function
$\hbbeta_n$ is then called a \emph{regularization path}. For instance we show
that the central limit theorem established for the lasso estimator in Knight
and Fu (2000) continues to hold in a functional sense for the regularization
path. Other examples include various possible contrast processes for $M_n$ such
as those considered in Pollard (1985).
