One-big-jump polynomial lower bound ($\mathbb{P}(S_n\ge na)\ge \mathbb{P}(X_1\ge na)$)

For nonnegative independent identically distributed summands with polynomial upper tail $\mathbb{P}(X_1\ge x)=x^{-\beta }$ for $x\ge 1$, positivity gives $\mathbb{P}(S_n\ge na)\ge (na{)}^{-\beta }$ 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.