Powers of Inner Products of Random Vectors, Uniformly Distributed on the Sphere

Our task in Chap. 4 is to determine the common denominator that underlies the real and complex case. To this end, we have to consider a common source of the Grothendieck inequality, which could be viewed as an “equality in mean”, induced by the classical Pearson correlation coefficient of the random signs of two suitably correlated Gaussians. We revisit uniform measures on spheres, Gaussian hypergeometric functions and even absolutely p-summing operators and reveal their connection with real and complex Gaussian variables, leading to the more general integration of expected powers of inner products of random vectors, uniformly distributed on the sphere, which are also of importance in statistical machine learning theory (in form of the so called “kernel trick”).