Dyadic Rademacher series ($Y={\sum }_{j\ge 1}{2}^{-j}{X}_j$)

For independent Rademacher random variables, this series converges absolutely on every sign sequence and has the uniform distribution on $[-1,1]$. Its first $n$ terms are uniform on the $2^n$ midpoints of equal subintervals of $[-1,1]$, giving weak convergence of probability measures by Riemann sums. The remaining tail has absolute value at most $2^{-n}$, and its mean square is $4^{-n}/3$ by independence. Thus convergence is also almost surely and in $L^2$.