Generalizations of the fourth moment theorem

. Azmoodeh et al. established a criterion regarding convergence of the second and other even moments of random variables in a Wiener chaos with fixed order guaranteeing the central convergence of the random variables. This was a major step in studies of the fourth moment theorem. In this paper, we provide further generalizations of the fourth moment theorem by building on their ideas. More precisely, further criteria implying central convergence are provided: (i) the convergence of the fourth and any other even moment, (ii) the convergence of the sixth and some other even moments.