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Infinite divisibility of a class of two-dimensional vectors with componentsin the second Wiener chaos is studied. Necessary and sufficient conditions forinfinite divisibility is presented as well as more easily verifiable sufficientconditions. The case where both components consist of a sum of two Gaussiansquares is treated in more depth, and it is conjectured that such vectors areinfinitely divisible.

On infinite divisibility of a class of two-dimensional vectors in the second Wiener chaos