 One-big-jump polynomial lower bound

Let $S_n={\sum }_{i=1}^n{X}_i$, where the summands are independent and identically distributed with Pareto distribution $\mathrm{Pareto}(\alpha ,\beta )$. Their positivity implies $\{X_1\ge na\}\subseteq \{S_n\ge na\}$, so for fixed $a>0$ and $na\ge \beta$,Both outer expressions tend to zero. A single large summand already prevents a strictly negative exponential rate. No positive exponential moment exists, so the exponential-moment form of the Cramér theorem cannot be applied. The bound proves this logarithmic rate and does not assert an exact tail asymptotic. In particular, it remains true when $a$ lies below a finite mean, where the event need not even be rare. The stronger fixed-summand-count asymptotic in a heavy-tailed subexponential distribution is a separate statement.