= Dyadic Rademacher series
{c}
{title2=$Y=\sum_{j\geq1}2^{-j}X_j$}
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 $L^2$.
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