Currently, I am working on a random series as follows. Let $\{Y_k\}$ be a sequence of i.i.d. Bernoulli random variables with expectation $p$. Then we define $$ S = \sum_{k=1}^\infty \prod_{\ell=1}^k 2^{2Y_\ell - 1}. $$ Now we assume that $0<p<1/2$. We can prove that $S<\infty$ a.s. Then the question is, what is the distribution function of this random series $S$? Let us use $F$ to denote its distribution function. Then, it is quite interesting to find that, $F$ satisfies the following functional equations for any $x\in\mathbb{R}$: $$ F(x) = p F(\frac{x}{2}-1) + (1-p) F(2x-1). $$ Furthermore, if we use $\varphi$ to denote the characteristic function of $F$, then we find that for any $t\in\mathbb{R}$, $$ \varphi(t) \;=\; p e^{2it} \varphi(2 t) + (1-p) e^{it/2} \varphi(t/ 2) \quad \text{for all } t \in \mathbb{R}. $$ I am interested in solving these functional equations or working on any special properties of $F$ or $\varphi$ (e.g., fractals or singularity). Any comment helps.
Following is an empirical distribution of a simple simulation with $p = 0.05$, where you can see self-similarity. I am curious whether this distribution is singular (w.r.t. Lebesgue measure).
