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pr.likelihood – Random Fourier succession in Hilbert house Answer

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pr.likelihood – Random Fourier succession in Hilbert house

Let $H$ breathe a Hilbert house, and $X_n$,
$ nin mathbb {Z}$, breathe a sequence of impartial 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 succession
$$
sum_n a_n X_n e^{int}, quad t in [0,2pi],
$$

is sort of certainly uniformly convergent in $H$?

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