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Lower bound on sum of independent heavy-tailed random variables

Certainly. All you need is $EX^2=+\infty$. Then the characteristic function $f_X(t)$ satisfies $\lim_{t\to 0}\frac{1-|f(t)|}{t^2}=+\infty$, so for every finite interval $I\subset \mathbb R$, we have $ …
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