A random variable whose digits in the $\widetilde {L}$-representation have the Markovian dependence

The distribution of the random variable \[ \theta =\frac {1}{\theta _1}+\sum ^{\infty }_{n=2}\frac {(-1)^{n-1}} {\theta _1(\theta _{1}+1)\dots \theta _{n-1}(\theta _{n-1}+1)\theta _{n}}\] is studied where $(\theta _n)$ is a homogeneous Markov chain assuming only positive integer values and having the initial distribution $(p_1, p_2,\dots , p_n,\dots )$ and transition matrix $\|p_{ik}\|$. The Lebesgue structure of the distribution (discrete, absolutely continuous, and singular components) is studied and topological, metric and fractal properties of the spectrum (the minimal closed support of the distribution) is investigated.