Before revision:  (the former scheme with some time dependence measure \cx^*(\ct)) or \log J /{T} = \Co(1) (the latter scheme with the mixing coefficient\beta_{mix=} \{(J^{2+\delta' \log J T)^{-1}\}, \delta' >0. }  We quantify the interplay between the estimators' consistency rate and the time dependence level, discuss an intuitive resampling scheme for threshold selection, and also prove a general cross-validation result justifying this.
After revision:  (the former scheme with some time dependence measure \cx^*(\ct)) or \log J /{T} = \Co(1) (the latter scheme with the mixing coefficient\beta_{mix=} \{(J^{2+\delta' \log J T)^{-1}\}, \delta' >0. } some upper bounded mixing coefficient). We quantify the interplay between the estimators' consistency rate and the time dependence level, discuss an intuitive resampling scheme for threshold selection, and also prove a general cross-validation result justifying this.