Let $S\_1,\dots, S\_n$ be Bernoulli random variables which are $4$-wise independent. We have that for each $i$, $P(S\_i = 1) = p$ for some fixed probability $0 < p < 1$. What can we say about $P(\forall i\; S\_i= 1)$ in terms of upper  and lower  bounds? 



Clearly $P(\forall i\; S\_i= 1) \geq p^n$ but what is the largest it can be?



Probability all Bernoulli random variables take value $1$ with limited independence
This has been treated in the literature:


<http://arxiv.org/pdf/0801.0059v3.pdf>


also see
<http://arxiv.org/pdf/1201.3261.pdf>


In particular, the upper bound goes to 0, but only polynomialy in $n$.


