= Solution
The transformation of the <uniform distribution> gives, for $x\geq1$,
$$
\mathbb P(X_1\geq x)=\mathbb P(U_1\leq x^{-2})=x^{-2}.
$$
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 bound>
$$
1\geq\mathbb P(W_n\geq na)\geq\mathbb P(X_1\geq na)=(na)^{-2}.
$$
Taking logarithms and dividing by $n$ gives an upper bound $0$ and a lower bound $-2\log(na)/n\to0$. Hence
$$
\boxed{\lim_{n\to\infty}\frac1n\log\mathbb P(W_n\geq na)=0\quad\text{for every }a\geq0.}
$$
This 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 $1$.
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