On modifications of the Lindeberg and Rotar' conditions in the central limit theorem

A modification of the Lindeberg and Rotar' conditions was considered in the papers by Presman and Formanov [Dokl. Math., 99 (2019), pp. 204–207] and [Dokl. Ross. Akad. Nauk Ser. Mat., 485 (2019), pp. 548–552 (in Russian)]. This modification was concerned with the sums of absolute (respectively, difference) moments of order $2+\alpha$ for the distributions of the summands truncated at the unit level. It was shown that, when checking the normal convergence, it is sufficient, instead of checking the convergence to zero of the Lindeberg or Rotar' characteristics for any $\varepsilon >0$, to check that there exists an $\alpha >0$ such that a characteristic (introduced in these papers) corresponding to this $\alpha$ converges to zero. Moreover, from the existence of such $\alpha$ it follows that the characteristic corresponding to any $\alpha >0$ also tends to zero. We show that the moment functions can be changed to more general functions and describe the class of such functions.