Let $(\mathcal A,\varphi)$ be a tracial noncommutative $*$-probability space, and fix $d \in \mathbb N$. For each $k \in \mathbb N$, let $a_k=(a_{k,1},\dots,a_{k,d})$ be a $d$-tuple of centered self-adjoint elements of $\mathcal A$, meaning that $a_{k,i}=a_{k,i}^*$ and $\varphi(a_{k,i})=0$ for every $1 \le i \le d$. Assume that the unital $*$-subalgebras $\mathcal A_k:=\operatorname{alg}(1,a_{k,1},\dots,a_{k,d})$ are freely independent and that the tuples $a_k$ are identically distributed, in the sense that $\varphi(a_{k,i_1}\cdots a_{k,i_m})=\varphi(a_{1,i_1}\cdots a_{1,i_m})$ for all $k,m \in \mathbb N$ and all indices $i_1,\dots,i_m \in \{1,\dots,d\}$.

For $n \in \mathbb N$ and $1 \le i \le d$, define $S_{n,i}:=n^{-1/2}\sum_{k=1}^{n} a_{k,i}$. Define the covariance matrix $C=(c_{ij})_{1 \le i,j \le d}$ by $c_{ij}:=\varphi(a_{1,i}a_{1,j})$. Then $C$ is real symmetric positive semidefinite. Consequently, there exists a tracial noncommutative $*$-probability space $(\mathcal B,\psi)$ and a self-adjoint semicircular system $(s_1,\dots,s_d)$ in $\mathcal B$ whose free cumulants satisfy $\kappa_1^{\psi}(s_i)=0$, $\kappa_2^{\psi}(s_i,s_j)=c_{ij}$, and $\kappa_m^{\psi}(s_{i_1},\dots,s_{i_m})=0$ for every $m \ge 3$ and every $i_1,\dots,i_m \in \{1,\dots,d\}$. The $d$-tuple $(S_{n,1},\dots,S_{n,d})$ converges in joint distribution to $(s_1,\dots,s_d)$. Equivalently, for every $m \in \mathbb N$ and every $i_1,\dots,i_m \in \{1,\dots,d\}$,

\begin{align*} \lim_{n \to \infty}\varphi(S_{n,i_1}\cdots S_{n,i_m})=\sum_{\pi \in NC_2(m)}\prod_{\{p,q\}\in \pi} c_{i_p i_q}, \end{align*}

where $NC_2(m)$ denotes the set of noncrossing pair partitions of $\{1,\dots,m\}$, with the convention that the sum is $0$ when $m$ is odd.