A probability measure which has Markov property (Q1046822)

\textit{M. Hata} and \textit{M. Yamaguti} [Japan J. Appl. Math. 1, 183--199 (1984; Zbl 0604.26004)] showed that the classical nowhere differentiable Takagi function \[ T(x) = \sum_{n=0}^\infty \frac{1}{2^n} d(2^n x), \quad \text{where } d(x) = \text{dist}(x,{\mathbb Z}), \] is the derivative of the distribution function \(L_r(x)\) of the binomial measure with respect to the parameter \(r\) (and taken at \(r=1/2\)). The functions \(L_r\) are well-known singular functions (except for \(r=1/2\)). In the present paper, the authors construct a family of probability measures which generalize the binomial measure. They compute the higher-order derivatives of their distribution functions with respect to the parameters and show that these coincide with a certain family of generalized Takagi functions. As an application, they prove an analog of the Trollope-Delange formula. The classical Trollope-Delange formula connects the average occurrence of the digit~``1'' in the binary expansion of the numbers \(1\dots n\) with the Takagi function. The authors' analog connects the average occurrence of each of the four possible blocks of length~2 in the binary expansion of the numbers \(1\dots n\) with their generalized Takagi functions.