The Rademacher functions are an explicit iid sequence with Bernoulli law. Does it exist an explicit construction of an iid sequence with uniform law?



Uniform iid sequence
$\newcommand\N{\mathbb N}$Let $(r\_k\colon k\in\N)$ be the sequence of the [Rademacher functions](https://en.wikipedia.org/wiki/Rademacher_system). Re-enumerate this sequence into a two-way array $(r\_{i,j}\colon(i,j)\in\N^2)$. So, the $r\_{i,j}$'s are iid random variables (r.v.'s, defined on the standard probability space over the interval $[0,1]$) each uniformly distributed on the two-point set $\{-1,1\}$.


For each $i\in\N$, define the r.v. $U\_i$ on the standard probability space over the interval $[0,1]$ by the formula
$$U\_i:=\sum\_{j=1}^\infty\frac{r\_{i,j}}{2^j}.$$
Then the $U\_i$'s are iid r.v.'s each uniformly distributed on $[-1,1]$.




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Moreover, if $F$ is any cumulative distribution function (cdf) and $X\_i:=F^{-1}(\frac{1+U\_i}2)$ for $i\in\N$, then the $X\_i$'s are iid r.v.'s each with cdf $F$. Here, as usual,
$$F^{-1}(u):=\inf\{x\in\mathbb R\colon F(x)\ge u\}$$
for $u\in(0,1)$.


So, one can explicitly get the distribution of any sequence of iid real-valued r.v.'s from the single sequence of the Rademacher functions.


