The concept of asymptotic normality takes on some new aspects when the dimensionality of the vector random variable under consideration is allowed to increase indefinitely. A necessary and sufficient condition for joint asymptotic normality in a new (strong) sense, in the case of independence, is given.
C. L. Mallows. "A Note on Asymptotic Joint Normality." Ann. Math. Statist. 43 (2) 508 - 515, April, 1972. https://doi.org/10.1214/aoms/1177692631