I have $n$ iid random variables $ X_i $, which satisfy $ X_i>0$. I want to know the relationship between the following two probabilities, $$\mathbb{P}\left(\sum_{i=1}^{n}X_i > z \right) \qquad \text{ and } \qquad \mathbb{P}\left(X_i > z\right) $$ Or equivalently, how to approximate $ \mathbb{P}\left(\sum_{i=1}^{n}X_i > z \right)$ using $ \mathbb{P}(X_i > z ) $ as tight as possible?

Thank you in advance!


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