processes, while the latter has extended the results of [@CCK2013] to a central limit theorem for hyperrectangles, or sparsely convex sets in more general. Extension of the CCK theory to statistics other than the sum of independent random vectors has also been studied in many articles: Weakening the independence assumption has been studied in e.g. [@ZW2015; @ZC2017; @CCK2014c; @CQYZ2018]; @Chen2017 and @CK2017 [@CK2017rand] have developed theories for $U$-statistics. Moreover, some authors have applied the CCK theory to statistical problems regarding high-frequency data; see @KK2017 and @Koike2017stein. Nevertheless, none of the above studies is applicable to our problem due to its non-ergodic nature. That is, the asymptotic covariance matrix is random and depends on the $\sigma$-filed of the original probability space, the asymptotic distribution is essentially non-Gaussian.

Meanwhile, inspection of the proofs of the CCK theory reveals that most the parts do not rely on *any* structure of the underlying statistics. To be precise, let $S_n$ be the random vector corresponding to the objective statistic and suppose that we aim at approximating the distribution of $S_n$ by its Gaussian analog $S_n^\dagger$ which has the same mean and covariance matrix as those of $S_n$. In the proofs of the CCK theory, the 