Limiting Curves for the Dyadic Odometer

Abstract

A limiting curve of a stationary process in discrete time was defined by E. Janvresse, T. de la Rue, and Y. Velenik as the uniform limit of the functions$$ t\mapsto \left(S\left(t{l}_n\right)- tS\left({l}_n\right)\right)/{R}_n\in C\left(\left[0,1\right]\right), $$ where S stands for the piecewise linear extension of the partial sum, Rn:= sup |S(tln) − tS(ln))|, and (ln) = (ln(ω)) is a suitable sequence of integers. We determine the limiting curves for the stationary sequence (f ∘ Tn(ω)) where T is the dyadic odometer on {0, 1}ℕ and $$ f\left(\left({\omega}_i\right)\right)=\sum \limits_{i\ge 0}{\omega}_i{q}^{i+1} $$ for 1/2 < |q| < 1. Namely, we prove that for a.e. ω there exists a sequence (ln(ω)) 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.