Limiting curves for the dyadic odometer

A limiting curve of a stationary process in discrete time was defined by É. Janvresse, T. de la Rue, and Y. Velenik as the uniform limit of the functions
$$ t\mapsto \big(S(tl_n) - tS(l_n)\big)/R_n \in C([0, 1]), $$
where $S$ stands for the piecewise linear extension of the partial sum, $R_n := \sup |S(tl_n) - tS(l_n))|$, and $(l_n) = (l_n(\omega))$ is a suitable sequence of integers. We determine the limiting curves for the stationary sequence $(f\circ T^n(\omega))$ where $T$ is the dyadic odometer on $\{0,1\}^{\mathbb{N}}$ and
$$f((\omega_i)) = \sum\limits_{i\geq 0}\omega_iq^{i+1}$$
for $1/2 < |q| < 1.$ Namely, we prove that for a.e. $\omega$ there exists a sequence $(l_n(\omega))$ such that the limiting curve exists and is equal to $(-1)$ times the Tagaki–Landsberg function with parameter $1/2q.$ The result can be obtained as a corollary of a generalization of the Trollope–Delange formula to the $q$-weighted case.