Let $H$ be a Hilbert space, and $X_n$, $ n\in \mathbb {Z}$, be a sequence of independent Bernoulli random variables $P(X_n = \pm 1) = \frac 12$. Is there a characterization of the sequences $a_n$, $ n\in \mathbb{Z}$, in $H$ such that the series $$ \sum_n a_n X_n e^{int}, \quad t \in [0,2\pi], $$ is almost surely uniformly convergent in $H$?

There is actually a full characterization for scalar sequences in the work of Marcus and Pisier (1981) via entropy integrals. The vector-valued case is an application of a deep result on Bernoulli processes (Bednorz-Latala, Annals of Mathematics 2013).

fullcharacterization (for the simple reason that it is still unknown) but there is a whole chapter on when the sum of a random Fourier series with scalar coefficients is almost surely continuous and the continuity is actually derived from the uniform convergence. $\endgroup$ – fedja Feb 27 at 16:30