N + aTt

The arguments below are based further on the prior assumption that the(θi/ d) are sampled uniformly on the hyper-sphere of radius1. Wewill assume further that these results can be extended in our setting withθi sampled from agaussian distribution. Notice that conditional on (θi)i sampled on the sphere of radius d, we have for alli {1,...,N} that ui Inthis section we complete the proof ofpropositions 2.0.1 and 2.0.2. So we get a pair of integro-differential equations in this case (wheras forGt(z) we had only one such equation). However, we have one additional differential equation in this case.