The central limit theorem for the product of sums of various random variables has been studied in a variety of settings. The purpose of this note is to show that this kind of result is a corollary of the invariance principle.
"A note on the invariance principle of the product of sums of random variables." Electron. Commun. Probab. 12 51 - 56, 2007. https://doi.org/10.1214/ECP.v12-1255