The fourth-moment theorem is extended to sums of two multiple integrals with different parity, with a counterexample for the same-parity case.
The fourth moment theorem on the Poisson space,
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Fourth-Moment Theorems for Sums of Multiple Integrals
The fourth-moment theorem is extended to sums of two multiple integrals with different parity, with a counterexample for the same-parity case.