It is a folklore that the summation of a series after randomization enjoys improved regularity. For analytic functions, this subject was initiated by Littlewood, Paley, Zygmund, etc. in the 1920's. It remained active for more than half century, but was largely dormant in the last few decades. Our recently renewed interest in this subject is connected with an effort to develop random operator theory. So in this talk we focus on the big three: random Hardy functions, random Bergman functions, and random Dirichlet functions.