structural:reformatting
Lacunary trigonometric series

Theorem (Salem – Zygmund) — Let U be a random variable distributed uniformly on (0,2π), and X = r cos(n U + a), where

Then X 1 + ⋯ + X k r 1 2 + ⋯ + r k 2 {\displaystyle {\frac {X_{1}+\cdots +X_{k}}{\sqrt {r_{1}^{2}+\cdots +r_{k}^{2}}}}} converges in distribution to N (0, 1 2) {\textstyle {\mathcal {N}}{\big (}0,{\frac {1}{2}}{\big)}}.