Random Fourier series in Hilbert space

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 $$H$$?

• Did you look in J.-P. Kahane’s book, Some Random series of functions. I believe this will contain answers to your question. – Anthony Quas Feb 23 at 7:40
• As far as I can tell, there's nothing in Kahane's book about that, not even a characterization of scalar sequences. – Dang Zheng Feb 27 at 8:01
• @DangZheng That's partially wrong: there is no full characterization (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. – fedja Feb 27 at 16:30