The non-linear limit theorem for ordinal numbers in Pascal automorphism

Let $x=\{x_n,n\in\textbf N\}\in\textbf Z_2$ be a point, and let $t_n(x)$ be the ordinal number of its length-$n$ initial fragment among the binary sequences of length $n$ having the same number of zeros. Let $m$ denote Haar measure on $\textbf Z_2$. Non-linear limit theorem. There exist a sequence $a_n$ of natural numbers and a sequence $b_n$ of positive numbers such that, for almost all $x\in\textbf Z_2$ with respect to Haar measure, there exists the limit

$$\lim_n m\left\{x:\frac{t_n(x)-a_n}{b_n}\leq\alpha\right\}=\Psi(\alpha),$$

where $\Psi$ is a non-degenerate distribution on $\textbf R$. This is presented as an open problem concerning the asymptotic distribution of ordinal positions in the Pascal automorphism; the supplied text does not establish the existence of the sequences or the limiting distribution.