= One-big-jump polynomial lower bound
{title2=$\mathbb P(S_n\geq na)\geq\mathbb P(X_1\geq na)$}
For nonnegative <independent> identically distributed summands with polynomial upper tail $\mathbb P(X_1\geq x)=x^{-\beta}$ for $x\geq1$, positivity gives $\mathbb P(S_n\geq na)\geq(na)^{-\beta}$ for fixed $a>0$ and large $n$. Since this <probability> is also at most one, its logarithm divided by $n$ tends to zero. No positive exponential moment exists, so the exponential-moment form of the <Cramér theorem> cannot be applied. The bound proves a logarithmic rate and does not assert an exact tail asymptotic.
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