Weak Convergence of P-statistics

In this chapter we shall prove a result on the weak convergence of generalized permanents which extends the results on random permanents presented in Chapter 3. Herein we return to the discussions of Chapter 1 where we introduced a matrix permanent function as a way to describe properties of perfect matchings in bipartite graphs. Consequently, the asymptotic properties which will be developed in this chapter for P-statistics can be immediately translated into the language of the graph theory as the properties of matchings in some bipartite random graphs. This will be done in Section 5.3, where we shall revisit some of the examples introduced in Chapter 1. In order to establish the main results of this chapter we will explore the path connecting the asymptotic behavior of U- and P-statistics. An important mathematical object which will be encountered here is a class of real random variables known as multiple Wiener-Itô integrals. The concept of the Wiener-Itô integral is related to that of a stochastic integral with respect to martingales introduced in Chapter 4, though its definition adopted in this chapter is somewhat different - it uses Hermité polynomial representations. It will be introduced in the next section. We shall start our discussion of asymptotics for P-statistics by first introducing the classical result for U-statistics with fixed kernel due to Dynkin and Mandelbaum, then obtaining a limit theorem for U-statistics with kernels of increasing order, and finally extending the latter to P-statistics.