We derive the optimal convergence rate in the central
limit theorem for the number of maxima of random samples chosen
uniformly at random from the right triangle of the shape

.
A local limit theorem with rate is also derived. The result is then
applied to the number of maxima in general planar regions
(upper-bounded by some smooth decreasing curves) for which a
near-optimal convergence rate to normality is established.

Please send comments and corrections to Thomas Klausner.