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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!

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$\mathsf{P}\{X_1 = 0, \dots, X_n = 0\} \le \min_i \mathsf{P} \{X_i = 0\} = 1 - \max_i p_i \le 1 - N^{-1/2} \Vert p \Vert_2$

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