Let's define a random variable $\displaystyle \chi$ as...

$\displaystyle \displaystyle \chi= \sum_{n=1}^{\infty} \chi_{n} = \sum_{n=1}^{\infty} \frac{z_{n}}{n}$ (1)

... where the $\displaystyle z_{n}$ are discrete random variables with $\displaystyle P \{z_{n}=-1\}= P \{z_{n}=-1 \}= \frac{1}{2}$. Each $\displaystyle \chi_{n}$ has p.d.f. given by...

$\displaystyle \displaystyle \sigma_{n}(x)= \frac{\delta(x -\frac{1}{n}) + \delta(x+ \frac{1}{n})}{2}$ (2)

... and each $\displaystyle \sigma_{n} (*)$ has Fourier transform given by...

$\displaystyle \displaystyle \Sigma_{n} (\omega) = \cosh (i\ \frac{\omega}{n}) = \cos \frac{\omega}{n}$ (3)

Setting $\displaystyle \sigma(x)$ the p.d.f of $\displaystyle \chi$ and $\displaystyle \Sigma(\omega)$ its Fourier transform is...

$\displaystyle \displaystyle \Sigma(\omega)= \prod_{n=1}^{\infty} \Sigma_{n} (\omega) = \prod_{n=1}^{\infty} \cos \frac{\omega}{n} $ (4)

Now $\displaystyle \sigma(x)$ [if it exists...] can be obtained as inverse Fourier Transform of $\displaystyle \Sigma(\omega)$... but that requires some more efforts from me!

...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$