Gaussian tails force an unbounded normalized partial-sum limsup (${\mathrm{limsup}}_n{S}_n/\sqrt{n}=+\infty \text{almost surely}$)

For independent identically distributed mean-zero variables with finite positive variance, the central limit theorem gives a positive eventual lower bound for $\mathbb{P}(S_n/\sqrt{n}\ge K)$ for every finite $K$. The decreasing events ${\bigcup }_{n\ge m}\{S_n/\sqrt{n}\ge K\}$ have probabilities bounded below, so continuity from above of a measure gives positive probability that the limsup is at least $K$. The limsup itself is unchanged by removing any finite initial sum, hence is tail measurable. Kolmogorov zero-one law makes each event “limsup at least $K$” certain; intersect over integer $K$. The result is false for zero variance. One should apply the zero-one law to the limsup, rather than assume the exact threshold infinitely-often event is unaffected by a vanishing change.