Multiple stochastic integrals with dependent integrators

Let [mu] be a [sigma]-finite measure, R = (rij) be a covariance matrix, and B1,..., Bn be dependent Gaussian measures satisfying EBi(A1) Bj(A2) = rij[mu](A1 [down curve] A2). Multiple integrals of the form In(f) = [integral operator]f(x1,..., xn) dB1(x1) ... dBn(xn), with f [set membership, variant] L2([mu]n) are investigated. A diagram formula is established and a class of functions which play the role of the Hermite polynomials for these more general integrals is introduced. Cumulants of double integrals are evaluated and the following result is established: if {Xj} and {Yj} are correlated stationary sequences of strongly dependent Gaussian random variables, then [Sigma]j=1[Nt] XjYj, adequately normalized, converges in D[0, 1] to I2(fi).