Chapter 11 - Limit theorems for generalized U-statistics

Summary

In this chapter we use the theory of Gaussian Hilbert spaces to obtain the asymptotic distributions of some important random variables. In the first section, we study U-statistics. In the second section we extend the results to asymmetric statistics. In the third section we extend the results further; as a special case we obtain results for random graphs.

Note that the original variables are defined without any reference to normal variables or Gaussian Hilbert spaces; the Gaussian Hilbert space is introduced as a convenient tool to treat the asymptotic distribution.

A common theme in these results is that ‘typically’ the asymptotic distribution is normal, but in some ‘degenerate’ cases other limits occur; these other limits can be represented as variables in a Wiener chaos H^:n: for some Gaussian Hilbert space H, and can for example be expressed as multiple Gaussian stochastic integrals. The explanation for this phenomenon that emerges from the proofs below is that the variable in question may be expanded as a sum, where each term converges in distribution to some chaos, and the first term is asymptotically normal. Typically, the first term dominates the sum and all others are asymptotically negligible, but in degenerate cases the first term vanishes and the sum is dominated by one or several later terms, which may converge to a higher order chaos.