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 $a_n:= (x_n + x_n^{-1})/\sqrt{2}$, where $\{x_i\}_{i\geq 0}$ represents the standard family of infinite generators. Analogously to the case of $F=F_2$, it is established that the limit distribution of $s_n = (a_0 + \ldots + a_{n-1})/\sqrt{n}$ 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.