Abstract
Consider a mean zero random variable $X$, and an independent sequence $(X_n)$ distributed like $X$. We show that the random Fourier series $\sum_{n\geq 1} n^{-1} X_n \exp(2i\pi nt)$ converges uniformly almost surely if and only if $E(|X|\log\log(\max(e^e, |X|))) < \infty$.
Citation
Michel Talagrand. "A Borderline Random Fourier Series." Ann. Probab. 23 (2) 776 - 785, April, 1995. https://doi.org/10.1214/aop/1176988289
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