Let $(\mathcal{A}, \varphi)$ be a noncommutative probability space, and let $(a_k)_{k \geq 1}$ be a freely independent, identically distributed sequence in $\mathcal{A}$ such that every $a_k$ has moments of all orders. Assume the variables are centered, meaning $\varphi(a_k) = 0$ for every $k \geq 1$. For each $n \in \mathbb{N}$, define

\begin{align*} S_n = \frac{1}{\sqrt{n}}\sum_{k=1}^{n} a_k. \end{align*}

Let $\kappa_m: \mathcal{A}^m \to \mathbb{C}$ denote the $m$-th free cumulant functional. Then, for every $n \in \mathbb{N}$,

\begin{align*} \kappa_1(S_n) = 0 \end{align*}

and

\begin{align*} \kappa_2(S_n, S_n) = \kappa_2(a_1, a_1). \end{align*}

Moreover, for every integer $m \geq 3$,

\begin{align*} \lim_{n \to \infty} \kappa_m(S_n, \dots, S_n) = 0. \end{align*}