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The transformation of the uniform distribution gives, for $x\ge 1$,Thus $X_1$ has a Pareto distribution with lower endpoint $1$ and shape $2$. It has mean $2$, but no positive exponential moments, so the preceding version of the Cramér theorem is unavailable.

At $a=0$, the event in question is certain. For every fixed $a>0$ and all sufficiently large $n$, positivity of the summands gives the one-big-jump polynomial lower boundTaking logarithms and dividing by $n$ gives an upper bound $0$ and a lower bound $-2\log (na)/n\to 0$. HenceThis covers thresholds below, at and above the mean. Above the mean it means decay is slower than exponential speed $n$, not that the tail probability tends to $1$.