# Uniform iid sequence

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

$$\newcommand\N{\mathbb N}$$Let $$(r_k\colon k\in\N)$$ be the sequence of the Rademacher functions. 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 $$[-1,1]$$.
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 $$u\in(0,1)$$.