Let $(\mathcal A,\varphi)$ be a noncommutative probability space, where $\mathcal A$ is a unital complex algebra and $\varphi:\mathcal A\to\mathbb C$ is a unital linear functional. Let $a_1,a_2,\dots \in \mathcal A$ be freely independent, identically distributed self-adjoint elements. Assume that $\varphi(a_i)=0$ and $\varphi(a_i^2)=1$ for every $i\geq 1$. Let $\kappa_r:\mathcal A^r\to\mathbb C$, for $r\geq 1$, denote the free cumulants associated to $\varphi$, and assume the standard cumulant characterization of freeness: mixed cumulants in variables from distinct free subalgebras vanish.

For each $n\geq 1$, define

\begin{align*} S_n:=\frac{a_1+\cdots+a_n}{\sqrt n}. \end{align*}

Then the sequence $(S_n)_{n\geq 1}$ converges in moments to the standard semicircular moment sequence. Explicitly, for every integer $m\geq 1$,

\begin{align*} \lim_{n\to\infty}\varphi(S_n^m)=\#NC_2(m), \end{align*}

where $NC_2(m)$ denotes the set of noncrossing pair partitions of $\{1,\dots,m\}$. In particular this limit is $0$ when $m$ is odd and is the Catalan number $\frac{1}{k+1}\binom{2k}{k}$ when $m=2k$.