On Natural Convergence Rate Estimates in the Lindeberg Theorem

We prove two estimates of the rate of convergence in the Lindeberg theorem, involving algebraic truncated third-order moments and the classical Lindeberg fraction, which generalize a series of inequalities due to Esseen (Arkiv För Matematik8, 1, 7–15, 1969), Rozovskii (Bulletin of Leningrad University (in Russian), (1):70–75, 1974), & Wang and Ahmad (Sankhya A: Indian J.Stat.78, 2, 180–187, 2016), some of our recent results in Gabdullin, Makarenko, Shevtsova, (J. Math Sci.234, 6, 847–885, 2018, J. Math Sci.237, 5, 646–651, 2019b) and, up to constant factors, also Katz (Ann. Math. Statist.34, 1107–1108, 1963), Petrov (Soviet Math. Dokl.6, 5, 242–244, 1965), & Ibragimov and Osipov (Theory Probab. Appl.11, 1, 141–143, 1966b). The technique used in the proof is completely different from that in Wang and Ahmad (Sankhya A: Indian J.Stat.78, 2, 180–187, 2016) and is based on some extremal properties of introduced fractions which has not been noted in Katz (Ann. Math. Statist.34, 1107–1108, 1963), Petrov (Soviet Math. Dokl.6, 5, 242–244, 1965), & Wang and Ahmad (Sankhya A: Indian J.Stat.78, 2, 180–187, 2016).