An extension of Krishnan's central limit theorem to the Brown-Thompson groups

We extend a central limit theorem, recently established for the Thompson group $F=F_2$ by Krishnan, to the Brown-Thompson groups $F_p$, where $p$ is any integer greater than or equal to $2$. The non-commutative probability space considered is the group algebra $\mathbb{C}[F_p]$, equipped with the canonical trace. The random variables in question are $a_n:=(x_n+x_n^{-1})\mathrm{/}\sqrt{2}$, where $\{x_i{\}}_{i\ge 0}$ represents the standard family of infinite generators. Analogously to the case of $F=F_2$, it is established that the limit distribution of $s_n=(a_0+\dots +a_{n-1})\mathrm{/}\sqrt{n}$ converges to the standard normal distribution. Furthermore, it is demonstrated that for a state corresponding to Jones’s oriented subgroup $\vec{F}$, such a central limit theorem does not hold.