Invariance principles for stochastic area and related stochastic integrals

Given an antisymmetric kernel K (K(z, z') = -K(z', z)) and i.i.d. random variates Zn, n[greater-or-equal, slanted]1, such that EK2(Z1, Z2)<[infinity], set An = [summation operator]1[less-than-or-equals, slant]i[less-than-or-equals, slant]j[less-than-or-equals, slant]n K(Zi,Zj), n[greater-or-equal, slanted]1. If the Zn's are two-dimensional and K is the determinant function, An is a discrete analogue of Paul Lévy's so-called stochastic area. Using a general functional central limit theorem for stochastic integrals, we obtain limit theorems for the An's which mirror the corresponding results for the symmetric kernels that figure in theory of U-statistics.