CDF of sum of independent cosines?

Question

Consider the random variable $$X=\frac{1}{d}\sum_{k=1}^d\cos X_k$$ where $X_k$ are each drawn uniformly i.i.d. from $[0,2\pi]$. What is the CDF of X?

It seems that a relatively direct way could be to find the PDF (and then integrate), but this seems difficult. Another approach is to recognize $\cos X_k=\Re\exp(i X_k)$, $\exp(i X_k)$ being drawn uniformly i.i.d. from $S^1\subset\mathbb{C}$, and so we are actually asking about the PDF/CDF of $\Re((S^1)^{\star d})$, $\star$ denoting convolution. However, both of these approaches seem rather difficult and I have not gotten very far with either.

For the $S^1$ approach, it seems that it might be fruitful to recognize the distribution on $S^1$ as a radial measure induced by the Dirac delta at $1$ in $\mathbb{R}_{\ge0}$, and to work more directly with radial measures when iterating. However this doesn't seem to get very far either.

It seems that these are natural enough questions that there might be some standard reference, though all I have found in the way of explicit distributions is this stats.SE answer which argues that asymptotically in $d$, $X$ will look like $N(0,\frac{1}{2d})$. This asymptotic case is relevant to me, but I am also curious about explicit, finite $d$.

Answer 1

The probability distribution density function of $Y=\cos X$ is $P(y)=(1/\pi)(1-y^2)^{-1/2}$, for $|y|<1$, with moment generating characteristic function $F(s)=J_0(s)$, so the moment generating characteristic function of $Z=\sum_{k=1}^d \cos X_k$ is $$F_d(s)=[J_0(|s|)]^d.$$ Transforming back to the probability distribution density function of $Z$, $$P_d(z)=\frac{1}{\pi}\int_0^\infty [J_0(s)]^d \cos(zs)\,ds.$$ I do not know of a closed form expression for this integral. The large-$d$ behavior is the Gaussian noted in the OP. For $d=2$ one has $$P_2(z)=\frac{2 }{\pi ^2 |z|}K\left(1-\frac{4}{z^2}\right),$$ with $K$ an elliptic integral.

The shapes for small $d$ are quite varied, here are plots of $P_d(z)$ for $d=1,2,3,4,5$ (from left to right).