= Random-walk maximum limit from Donsker invariance
{title2=$n^{-1/2}\max_{j\leq n}S_j\Rightarrow\sup_{s\leq1}B_s$}
For a <random walk> with centered <independent> identically distributed increments of <variance> one, the maximum of its polygonal interpolation is the maximum at its grid vertices. The maximum functional is Lipschitz for the <supremum norm>, so the <continuous mapping theorem> applied to the <Donsker invariance principle> proves the displayed <convergence in distribution>. The <Brownian reflection principle> then gives upper-tail limits $2(1-\Phi(a))$ at all $a>0$, since the limiting maximum has no atom there.
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