Central Limit Theorem by moments

In a previous Central Limit Theorem by moments, it has been proved that the moments converge to those of the normal distribution if the moments of sums are asymptotically independent (cf. Blacher, R., 1990. Theoreme de la limite centrale par les moments. C. R. Acad. Sci. Paris. 311(I), 465-468). In this paper we generalize this result by adding a negligible sequence to these sums. So, we can prove that the moments of some functionals of strong mixing sequences converge.