has been given in Theorem 5.1 of @NNP2016. However, this result is not directly applicable to the current situation because it assumes that the components of $S_n^\dagger$ are conditionally independent, which is less interesting to statistical applications (and especially not the case in the problem illustrated above). To remove such a restriction from the result of [@NNP2016], we employ the novel interpolation method introduced in @NY2017, instead of Slepian’s interpolation used in [@NNP2016] and the original CCK theory. Another problem in the present context is validation of standardizing statistics by random variables. In a low-dimensional setting, this is typically achieved by proving the so-called *stable convergence in law* (see e.g. [@PV2010] for details). However, in a high-dimensional setting, the meaning of stable convergence is unclear and its naïve extension is not useful because of the lack of the continuous mapping theorem and the delta method (see Section \[sec:main\] for a relevant discussion). So we also aim at developing a formulation appropriate to validating such an operation. The remainder of the paper is organized as follows. Section \[sec:preliminaries\] is devoted to some preliminaries on notation and concepts used in the paper. Section \[sec:main\] presents the main results obtained in this paper. 