On the functional limits for sums of a function of partial sums

We derive a functional central limit theorem (fclt) for normalised sums of a function of the partial sums of independent and identically distributed random variables. In particular, we show, using a technique presented in Huang and Zhang [Huang, W., Zhang, L.-X., 2007. A note on the invariance principle of the product of sums of random variables. Electron. Comm. Probab. 12, 51-56] that the result from Qi [Qi, Y., 2003. Limit distributions for products of sums. Statist. Probab. Lett. 62, 93-100], for normalised products of partial sums, can be generalised in this fashion to a fclt.