For a random vector $\xi_n \in \mathbb{R}^p$, if $\xi_n \rightarrow_d N(\mu, \Sigma)$, we can construct \begin{equation*} \Psi := \xi_n^{\top} \widehat{\Sigma}^{-1} \xi_n \end{equation*} for testing whether $\mu = 0$, where $\widehat{\Sigma}$ is an reasonable estimate of $\Sigma$.

Now assume $\xi_n$ converge to a mixed normal distribution, i.e. \begin{equation*} \xi_n \rightarrow_d \sum_{k = 1}^K w_k Z_k \end{equation*} where $w_1 +\cdots + w_K = 1$ and $Z_1, \cdots, Z_K$ are independently distributed as $N(0, \Sigma_k)$. Since the elements in one $\Sigma_k$ are quite different from those in the other, we cannot write this mixed normal distribution as a single normal distribution. In this case, I try to construct $$\tilde{\Psi} = \sum_{k = 1}^K w_k \xi_n^{\top} \widehat{\Sigma}_k^{-1} \xi_n$$ as the statistic. However, it seems to be difficult to analyze the properties of this statistic since the components on the right side are NOT independent from each other.

Do anyone know how to construct a proper statistic in this case. Thanks so much!