In this paper we show that a necessary and sufficient condition on a Banach space $B$ for the validity of the accompanying laws theorem is that $c_0$ is not finitely representable in $B$ or, equivalently, that $B$ is of cotype $q$ for some $q > 0$. The proof is based on a result of Maurey and Pisier on the geometry of these spaces and on a theorem about approximation in $L_p$ of Banach valued triangular arrays by finite dimensional ones.
"On the Accompanying Laws Theorem in Banach Spaces." Ann. Probab. 9 (2) 202 - 210, April, 1981. https://doi.org/10.1214/aop/1176994462