Asymptotics for sums of a function of normalized independent sums

We derive a central limit theorem for sums of a function of independent sums of independent and identically distributed random variables. In particular, we show that previously known result from Rempała and Wesołowski [Rempała, G., Wesolowski, J., 2005. Asymptotics for products of independent sums with an application to Wishart determinants. Statist. Probab. Lett. 74 129-138], which can be obtained by applying the logarithm as the function, holds true under weaker assumptions.