# pr.probability – Random Fourier series in Hilbert space

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