= Brownian fluctuations exceed the square-root scale
{c}
{title2=$\limsup_{t\to\infty}B_t/\sqrt t=+\infty$}
At integer times, $B_n/\sqrt n$ has the same standard <normal distribution> for every $n$. The event that its limit superior exceeds a fixed finite level is a <tail event> of the independent unit-time Brownian increments, since changing any finite initial segment contributes only a term tending to zero. The <Kolmogorov zero-one law> applies. Its probability is positive because each fixed-time exceedance has the same positive probability, giving a positive probability of infinitely many exceedances by the decreasing union-of-tail-events argument. Thus its probability is one. Intersecting over integer levels proves the stated divergence, without the lower iterated-logarithm bound.
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