Infinite divisibility of a class of two-dimensional vectors with components in the second Wiener chaos is studied. Necessary and sufficient conditions for infinite divisibility are presented as well as more easily verifiable sufficient conditions. The case where both components consist of a sum of two Gaussian squares is treated in more depth, and it is conjectured that such vectors are infinitely divisible.
In this paper we define the consistent criteria of hypotheses such as the probability of any kind of errors is zero for given criteria. We prove necessary and sufficient conditions for the existence of such criteria.