INDEPENDENCE OF DOUBLE WIENER INTEGRALS
给出了两个双重维纳积分统计独立的充要条件，并证明了对多个积分而言两两独立蕴含相互独立，同时将结果推广到不连续核，涵盖了正态变量二次型独立性的经典定理。
In this paper a necessary and sufficient condition is obtained for two double Wiener integrals to be statistically independent, first in the case of symmetric and continuous kernels. It is also shown that, for more than two double Wiener integrals, pairwise independence implies mutual independence. After that, the continuity condition on the kernels is somewhat relaxed, and it is shown that Craig's (1943, Annals of Mathematical Statistics 14, 195–197) theorem on the independence of quadratic forms in normal variables is a special case of the result obtained for the case of discontinuous kernels.