I would like a nice expression of my weird probability density function (in blue): it is a gaussian density (black dots) reflecting on itself between two mirrors.
It is defined on a bounded real support: $d: [f, c] \rightarrow \mathbb{R}^+$, between a floor $f$ and a ceiling $c$: $T=c-f$.
Consider a gaussian density $\displaystyle \varphi_0: t \mapsto \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(t-\mu_0)^2}{2\sigma^2}}$ with $\mu_0 = \mu \in [f, c]$. $\varphi_0$ cannot fit inside $T$ because it has infinite $\mathbb{R}$ support.
To fit inside the support, we consider its "reflections" on the two mirrors $f$ and $c$. If you stood on the support, you could see an infinite series of gaussian densities $(\varphi_i)_{i \in \mathbb{Z}}$, each parametrized by $\mu_i$, with:
$\displaystyle \mu_i = \left\{\begin{array}{1} i > 0: \left\{\begin{array}{2} i\ \text{even}:& f + i\, T + (\mu - f)\\ i \ \text{odd}:& f + i\, T + (c - \mu) \end{array}\right.\\ i < 0: \left\{\begin{array}{2} i\ \text{even}:& f + (i+1)\, T - (c - \mu)\\ i \ \text{odd}:& f + (i+1)\, T - (\mu - f) \end{array}\right.\end{array}\right., \mu_0 = \mu \in [f, c]$
In other words, $\displaystyle d(t) = \sum_{\mathbb{i \in Z}}\varphi_i(t)$, an infinite sum of gaussian densities parametrized by these $\mu_i$.
I know it must converge, since it is nothing but one gaussian sliced into flipped and unflipped pieces. But I would like a better expression of $d(t)$. Is there one?
For example, I have a feeling that it may be expressed in terms of the Jacobi theta function, but I cannot dig it out. Any idea?