The distributions of random incomplete sums of a series with positive terms satisfying the property of non-linear homogeneity

The Lebesgue type as well as topological, metric, and fractal properties of the spectrum of the distribution of the random variable \[ \xi =\sum _{n=1}^{\infty }a_n\xi _n \] are studied, where $\sum _{n=1}^{\infty }a_n=a_1+a_2+\dots +a_n+r_n$ is a convergent series with positive terms such that $r_{n+1}=a_{n+1}a_{n}$ for any $n\in \mathbb {N}$ and $(\xi _n)$ is a sequence of independent random variables taking only two values, $0$ and $1$, with probabilities $p_{0n}$ and $p_{1n}$, respectively. We describe the point spectrum in the discrete case, and we prove that the distribution of $\xi$ is of a Cantor singular type with an anomalous fractal spectrum in the continuous case. We also prove that the $n$-fold convolution of the random variable $\xi$ with itself has an anomalous fractal distribution.