= One-big-jump polynomial lower bound
{title2=$\mathbb P(S_n\geq na)\geq\mathbb P(X_1\geq na)$}
Let $S_n=\sum_{i=1}^nX_i$, where the summands are <independent and identically distributed> with <Pareto distribution> $\operatorname{Pareto}(\alpha,\beta)$. Their positivity implies $\{X_1\geq na\}\subseteq\{S_n\geq na\}$, so for fixed $a>0$ and $na\geq\beta$,
$$
\mathbb P(S_n\geq na)\geq\left(\frac{\beta}{na}\right)^\alpha,\qquad
\frac{\alpha}{n}\log\frac{\beta}{na}\leq\frac1n\log\mathbb P(S_n\geq na)\leq0.
$$
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.
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