Since the "unit" frequency differences (see (2.2) below) are dependent, the usual methods for establishing the normal character of the asymptotic distribution of the sum of random variables fail. However, the essential character of the distribution is disclosed by the integral functional relationship (3.6). From this it is possible to show that for large samples the distribution approximates "stability" in the normal sense ( and Lemma 2). Using the condition that the third logarithmic derivative of the characteristic function is uniformly bounded for all $n$ on a neighborhood of $t = 0$ one can prove that the asymptotic distribution exists and is normal.
"On the Asymptotic Distribution of the Sum of Powers of Unit Frequency Differences." Ann. Math. Statist. 21 (2) 263 - 271, June, 1950. https://doi.org/10.1214/aoms/1177729843