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 certain renormalization of the process. We determine the limiting curves for the stationary sequence (f ∘ Tn(ω)) where T is the dyadic odometer and f is the weighted sum of digits function. Namely, we prove that for a.e. ω there exists a sequence 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.