Multiple stochastic integrals with dependent integrators

Robert Fox and Murad S. Taqqu

Journal of Multivariate Analysis, 1987, vol. 21, issue 1, 105-127

Abstract: 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).

Keywords: limit; theorems; weak; convergence; multiple; integrals; long-range; dependence; multivariate; time; series; fractional; Gaussian; noises; Rosenblatt; process; Hermite; polynomials; diagram; formula; cumulants (search for similar items in EconPapers)
Date: 1987
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Citations: View citations in EconPapers (11)

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