Weak convergence of the regularization path in penalized M-estimation

We consider an estimator defined as the element minimizing a contrast process for each t. We give some general results for deriving the weak convergence of in the space of bounded functions, where, for each t, is the minimizing the limit of as . These results are applied in the context of penalized M-estimation, that is, when , where is a usual contrast process and a penalty such as the norm or the squared norm. The function 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 such as those considered in Pollard (1985).
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