If we are given iid samples $z_i$ taken uniformly with $z_i\in\Bbb Z$ with $|z_i|\in(2^n,2^{n+1})$ then at what $t$ we can expect $|\frac1t\sum_{i=1}^tz_i|<\epsilon$?
Does this help? Convergence Rate of Sample Average Estimator
So do we need $O(n)$ samples?
If we are given iid samples $z_i$ taken with $z_i\in\Bbb Z$ with $|z_i|\in(2^{n-m},2^{n+1})$ where the distribution is if $p(z_i)=w$ if $|z_i|\in(2^{n},2^{n+1})$ then $p(z_i)=w/2$ if $|z_i|\in(2^{n-1},2^{n})$ and $p(z_i)=w/4$ if $|z_i|\in(2^{n-2},2^{n-1})$ and so on then then at what $t$ we can expect $|\frac1t\sum_{i=1}^tz_i|<\epsilon$?
