I would like to know whether the following stochastic process is well studied.
Let $\{U_k: k \ge 1\}$ be a sequence of i.i.d random variable. $U_1$ is uniformly distributed on the unit interval $[0, 1]$. Now consider the stochastic process $(F(t))_{0<t<1}$, where
$$F(t)= \sum_{k = 1}^\infty 1_{U_k \le t^k}.$$
I would like to know the potentially existing literature on the above process. Does this process have a name ?
Any comment and reference is welcomed. Thanks a lot in advance.