I have a set of Bernoulli random variables $X\_1,X\_2,\ldots, X\_n$ and I would want to bound the joint probability $P(X\_1=0,X\_2=0,\ldots, X\_n=0)$ using the norm $\lvert \lvert \mathbf{p}\rvert \rvert$, where $$\mathbf{p}:=[P(X\_1=1)\quad P(X\_2=1)\ldots P(X\_n=1)]^\top$$


For the lower bound, I use Bonferroni's inequality: $P(\cap A\_i)\ge \sum P(A\_i)-N+1$ for events $\{A\_i\}$. Defining $A\_i:=\{X\_i=0\}$, 


\begin{align}
P(X\_1=0,X\_2=0,\ldots, X\_n=0)&\ge \sum P(X\_i=0)-N+1\\
1-P(X\_1=0,X\_2=0,\ldots, X\_n=0)&\le N-\sum P(X\_i=0)\\
&=\sum P(X\_i=1)\\
&=\mathbf{1}^\top \mathbf{p}\\
&\le \sqrt{N}\lvert\lvert \mathbf{p}\rvert\rvert
\end{align}


I wish to derive a similar bound on the other side, but I end up with trivial inequalities. Using $P(\cap A\_i)\le \sum P(A\_i)$ for events $\{A\_i\}$,


\begin{align}
P(X\_1=0,X\_2=0,\ldots, X\_n=0)&\le \sum P(X\_i=0)\\
1-P(X\_1=0,X\_2=0,\ldots, X\_n=0)&\ge 1-\sum P(X\_i=0)\\
&=1-N+\sum P(X\_i=1)\\
&=1-N+\lvert\lvert \mathbf{p}\rvert\rvert\_1\\ 
&\ge 1-N+\lvert\lvert \mathbf{p}\rvert\rvert\_2\\ 
\end{align}


Unfortunately the last bound is pretty trivial for probabilities. Can someone help me get a tight lower bound for $1-P(X\_1=0,X\_2=0,\ldots, X\_n=0)$? Thanks!



Relating joint probability to norm of vector of probabilities
$\mathsf{P}\{X\_1 = 0, \dots, X\_n = 0\} \le \min\_i \mathsf{P} \{X\_i = 0\} = 1 - \max\_i p\_i \le 1 - N^{-1/2} \Vert p \Vert\_2$


