In  a central limit theorem [CLT] for sequences of univariate random linear forms was proved. That result is extended in this note to a multivariate CLT for $q$-dimensional linear forms of constant $q$-vectors with real-valued random weight coefficients. Some applications are indicated in Section 3.
F. Eicker. "A Multivariate Central Limit Theorem for Random Linear Vector Forms." Ann. Math. Statist. 37 (6) 1825 - 1828, December, 1966. https://doi.org/10.1214/aoms/1177699175