Suppose that $B(t) : 0 \leq t < 1$ is the restriction of Brownian motion on the interval , hence a bounded function $f$ on $S^1$. It has some Fourier coefficients, $a_n = \int_{S^1} f(\theta) e^{-in \theta} d\theta$, which are random variables.
What is the distribution of these Fourier coefficients?
We can compute the mean of $a_n$: $E[a_n] = \int_{S^1} E(f(\theta)) e^{in\theta} = 0$, as $E(f(\theta)) = 0$ for each $\theta$. (I'm sloppily applying Fubini's theorem and crossing my fingers.)
Now I'm trying to compute the variance but I'm stuck. Does anyone have a suggestion?