Given $n$ numbers $x_i,i\in\{1,2,...,n\}$, define the set of all possible summation of $x_i$ with different signs: $$A = \Big\{\Big|\sum_i x_i s_i\Big| : s_i = \pm 1\Big\}$$ and define the minimum of $A$ as $y$.
Now let $x_i$ iid be sampled from the standard Gaussian $\mathcal{N}(0,1)$. It is expected that $y$ will go to zero when $n$ go to infinity. But how to prove it and how to estimate the rate? How to estimate the mean and variance of $y$ (as a random variable)?