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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.

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  • $\begingroup$ Wild guess: it's an inhomogeneous Poisson process, or a time-change of one. $\endgroup$ Nov 15, 2014 at 14:53
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    $\begingroup$ It seems that by computing the joint characteristic function of $F(t)$ and $F(t+s) - F(t)$, these two r.v.'s can not be independent. $\endgroup$
    – Yanqi QIU
    Nov 15, 2014 at 16:33

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