$\Xi_n\mathfrak{Z}_n$ while we replace $\mathfrak{C}_n,W_n$ and $\Xi_n$ by their estimators.

\[prop:comparison\] For each $n$, let ${\widehat{\mathfrak{C}}}_n,{\widehat{W}}_n$ and ${\widehat{\Xi}}_n$ be a $d\times d$ symmetric positive semidefinite random matrix, a $d$-dimensional random vector and an $m\times d$ random matrix, respectively. Set $\widehat{\mathfrak{Z}}_n:=\widehat{\mathfrak{C}}_n^{1/2}\zeta_n+{\widehat{W}}_n$. Suppose that $$\label{eq:consistent}
\sqrt{\log m}\|{\widehat{\Xi}}_n{\widehat{W}}_n-\Xi_nW_n\|_{\ell_\infty}\to^p0,\qquad
(\log m)^2\|{\widehat{\Xi}}_n\widehat{\mathfrak{C}}_n{\widehat{\Xi}}_n^\top-\Xi_n\mathfrak{C}_n\Xi_n^\top\|_{\ell_\infty}\to^p0$$ as $n\to\infty$. Then we have $$\sup_{y\in\mathbb{R}^m}|P({\widehat{\Xi}}_n\widehat{\mathfrak{Z}}_n\leq y|\mathcal{F})-P(\Xi_n\mathfrak{Z}_n\leq y|\mathcal{F})|\to0$$ as $n\to\infty$, provided that holds true.

We remark that the above proposition only gives a way to estimate the $\mathcal{F}$-conditional distribution of $\Xi_n\mathfrak{Z}_n$ when we have appropriate estimators for relevant variables: It says nothing about how to estimate the *unconditional* distribution of $\Xi_n\mathfrak{Z}_n$. Because of the non-ergodic nature of the problem, in general there seems no hope of consistently estimating the latter quantity even if we can consistently estimate unknown variables contained in $\Xi_n\mathfrak{Z}_n$. In a low-dimensional setting this issue is usually resolved by standardizing the objective statistic by a consistent estimator for its asymptotic covariance matrix, which is validated via the stability of convergence in law. In a high-dimensional setting, however, standardizing the (joint) distribution of the objective statistic is often difficult: Estimators for the conditional covariance matrix of the objective statistic are usually singular because the sample size is smaller than the dimension, 