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We derive a central limit theorem for sums of a function of independent sumsof independent and identically distributed random variables. In particular weshow that previously known result from Rempa\la and Weso\lowski (Statist.Probab. Lett. 74 (2005) 129--138), which can be obtained by applying thelogarithm as the function, holds true under weaker assumptions.

Asymptotics for sums of a function of normalized independent sums