Billard's property over the infinite‐dimensional polydisc

This paper is concerned with random holomorphic functions over the infinite‐dimensional polydisc. The celebrated Billard's theorem states, among other things, that for a random holomorphic function in a single variable—obtained by placing a random sign before each Taylor coefficient—the boundedness of the function can be automatically improved to continuity. We extend Billard's theorem in three aspects: first, we extend it to the case of infinitely many variables. Second, we establish an inverse result, that is, not merely a sufficient condition. Such results are rare in the literature. In particular, we define Billard's property (BP) for random variables and prove: (1) all symmetric random variables have BP; (2) a square‐integrable random variable has BP if and only if it is centered. Third, another novelty of our results is that they encompass applications to random Dirichlet series.