We present lower estimates for the best constant appearing in the weak
$(1,1)$ maximal inequality in the space $(\R^n,\|\cdot\|_{\iy})$. We show that
this constant grows to infinity faster than $(\log n)^{1-o(1)}$ when $n$ tends
to infinity. To this end, we follow and simplify the approach used by J.M.
Aldaz. The new part of the argument relies on Donsker's theorem identifying the
Brownian bridge as the limit object describing the statistical distribution of
the coordinates of a point randomly chosen in the unit cube $[0,1]^n$ ($n$
large).