To the talk page https://en.wikipedia.org/wiki/Draft_talk:Piet_Groeneboom I still want to add the following,
To point 2 of the account I want to add that in the 1988 paper in PTRF on convex hulls of samples in the plane by Piet Groeneboom also central limit theorems for the vertices of convex hulls of uniform samples in the interior of convex sets with smooth boundaries (circles, ellipses) were proved. The rate changes from sqrt(log n) to n^{1/3} in that case. This is also mentioned in Chapter 12 of the Handbook of Discrete and Computational Geometry (2017), written by Rolf Schneider. Concerning point 3: the property one needs in proving the conjecture of Perlman and Olkin ("power of the test is monotone in the eigenvalues of the non-centrality matrix"), following their approach is called "pairwise TP_2". One needs this if one wants to apply the FKG inequality. But unfortunately, the "pairwise TP_2" property does not hold, as shown in the Indagationes (2000) paper (an unchanged version of the 1983 MSRI report), so bad luck for the persons who tried it that way. Michael Perlman had a stack of these MSRI reports in his drawer, to send to people who sent him false proofs of this type.Petrusgr (talk) 08:10, 29 March 2023 (UTC)