= Gaussian tails force an unbounded normalized partial-sum limsup
{c}
{title2=$\limsup_n S_n/\sqrt n=+\infty\quad\text{almost surely}$}
For independent identically distributed mean-zero variables with finite positive variance, the <central limit theorem> gives a positive eventual lower bound for $\mathbb P(S_n/\sqrt n\geq K)$ for every finite $K$. The decreasing events $\bigcup_{n\geq m}\{S_n/\sqrt n\geq K\}$ 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.
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