Ustunel and Zakai have recently obtained a necessary and sufficient condition for two multiple Wiener integrals with respect to the same Brownian motion to be independent. In the present note, the sufficiency of their condition is shown to be a simple consequence of the classical relationship between multiple Wiener integrals and Hermite polynomials. The original proofs use Malliavin calculus.
Olav Kallenberg. "On an Independence Criterion for Multiple Wiener Integrals." Ann. Probab. 19 (2) 483 - 485, April, 1991. https://doi.org/10.1214/aop/1176990436