I have the problem where I am attempting to find the distribution of a random variable $Y$ which is defined by \begin{equation} Y = \sum_{n=1}^{N} X_{n} , \end{equation} where \begin{equation} X_{n} \sim Arcsine(-a_{n}, a_{n}), \quad n = 1, ..., N. \end{equation}
This problem arises when I have a set of known signals with deterministic (but unknown) amplitudes and random phases. That is, each $X_{n}$ is the result of \begin{equation} X = a_{n} \cos\Theta_{n}, \quad \Theta_{n} \sim U[-\pi,\pi). \end{equation}
Through experimentation in Matlab, lots of looking is Gradshteyn and Rhizhik, and finally finding Kausel and Baig [1], I think I have the expression for the distribution when $N = 2$.
That said, does the resulting distribution have a name? Does anyone know what the distribution is for $N$ terms? Of course if the $\{a_{n}\}$ are all equal, Central Limit Theorem applies and it will converge to a Gaussian, but I'm interested in fully characterizing things for the non-asymptotic case.