Is there any relationship between Rademacher distribution and Normal distribution? The Rademacher distribution is given as
The probability mass function of this distribution (https://en.wikipedia.org/wiki/Rademacher_distribution) is
${\displaystyle f(k)= \left\{{\begin{matrix}1/2&{\mbox{if }}k=-1,\\1/2&{\mbox{if }}k=+1,\\0&{\mbox{otherwise.}}\end{matrix}}\right.}$
In terms of the Dirac delta function, as
${\displaystyle f(k)={\frac {1}{2}}\left(\delta \left(k-1\right)+\delta \left(k+1\right)\right).} $
Agree with the comments on "relationship between the distributions". But what I am saying is you can approximate the sum of Rademacher random variables by a Normal distribution.
If random variable X has a Rademacher distribution, then $\frac{X+1}{2} $has a $Bernoulli(1/2)$ distribution.
Sum of i.i.d $Bernoulli(1/2)$ random variables follows a Binomial distribution with parameters $n$ and $p = 1/2$, where $n$ is the number of trials. So, for sufficiently large $n$ and a given $p $, the Binomial distribution can be approximated by Normal distribution, i.e. ${\displaystyle {\mathcal {N}}(np,\,np(1-p))}$.