for every $j$. Here, we assume that the dimension $d$ possibly depends on $n$ as $d=d_n$, while $q_j$’s do not depend on $n$. We also assume $d_n\geq{\textcolor{black}{3}}$ for every $n$ and ${\overline{q}}:=\sup_{j}q_j<\infty$. Our aim is to study mixed-normal limit theorems for the following functionals: $$Z_n=M_n+W_n,\qquad n=1,2,\dots,$$ where $W_n$’s are $d$-dimensional random vectors which represent the uncentered part of the functionals. Let us introduce mixed-normal random vectors approximating the functionals $Z_n$ in law as follows: $$\mathfrak{Z}_n=\mathfrak{C}_n^{1/2}\zeta_n+W_n,\qquad n=1,2,\dots.$$ Here, $\mathfrak{C}_n$ is a $d\times d$ symmetric positive semidefinite random matrix and $\zeta_n$ is a $d$-dimensional standard Gaussian vector independent of $\mathcal{F}$, which is defined on an extension of the probability space $(\Omega,\mathcal{F},P)$ if necessary. The main aim of this paper is to investigate reasonable regularity conditions under which the distribution of $Z_n$ is well-approximated by that of $\mathfrak{Z}_n$. To be precise, we are interested in the following type of result: $$\sup_{z\in\mathbb{R}^d}\left|P(Z_n\leq z)-P(\mathfrak{Z}_n\leq z)\right|\to0\qquad\text{as }n\to\infty.$$ It is well-recognized in statistic literature, however, that this type of result is usually insufficient for statistical applications because it does not ensure standardization by a *random* vector which is still random in the limit; such an operation is crucial for Studentization in the present context. In a low-dimensional 