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Article

Предельные кривые для диадического одометра

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 \[t\mapsto \big(S(tl_n) - tS(l_n)\big)/R_n \in C([0, 1]),\] where $S$ refers to 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 $f((\omega_i)) = \sum_{i\geq 0}\omega_iq^{i+1},$ for $1/2 < |q| < 1.$ That is, for a.s. $\omega$, there 
exist $(l_n(\omega))$ such that the limiting curve exists and is equal to $(-1)$ times the Tagaki-Landsberg function with parameter $1/2q.$