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Let $X := {1\dots m}$ be the set of indexes corresponding to the elements of the vector of i.i.d. random variables $w:=(w_1, \dots, w_m)$. Let there be the subsets $Y, Z \subseteq X$ which may or may not overlap.

I am trying to obtain a lower bound on the expression

$\mathbb{P}\left(|\prod_{i \in Y} w_i|\geq 1 + |\sum_{i \in Z}w_i)|\right)$.

For example, for random variables $w:=(a, b, c)$, I would be interested e.g. in a lower bound for

$\mathbb{P}(|ab| \geq 1 + |b+c|)$.

Perhaps there is a concentration inequality, transformation, etc. that applies to this case?

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    $\begingroup$ Your expression and your for example don't quite match up -- did you mean for your for example to be $|ab| \geq 1 + |b + c|$? $\endgroup$
    – Aaron Montgomery
    Commented Jun 17, 2022 at 18:14
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    $\begingroup$ exactly, just fixed it. Thank you! $\endgroup$
    – Scriddie
    Commented Jun 17, 2022 at 18:48
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    $\begingroup$ This is an interesting question. I suppose such a bound (if it exists) would need to depend on the support of the variables; for instance, if your variables are supported on $[0, 1]$, then the probability of this happening is $0$, so your bound would need to be trivial. $\endgroup$
    – Aaron Montgomery
    Commented Jun 17, 2022 at 18:50
  • $\begingroup$ Agreed, and that would probably include the degree of symmetry around zero (with complete symmetry perhaps being the worst case due to cancellations in the sum). $\endgroup$
    – Scriddie
    Commented Jun 17, 2022 at 19:53

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