Moment convergence in regularized estimations

In this paper we study uniform tail-probability estimates in regularized estimation, by making use of the polynomial type large deviation inequality for the associated statistical random fields. The statistical random fields in our model setup may be of mixed-rates type, may not be locally asymptotically quadratic. Our results provide a measure of rate of consistency in variable selection in sparse estimation, and also enable us to verify various arguments requiring convergence of moments of estimator-dependent statistics, such as the expected maximum-likelihood for AIC-type model criteria.