On the necessity of Statulevičius' condition in limit theorems for large-deviation probabilities (Q1589831)

This paper considers the problem of necessity in some sense of Statulevicius' condition (SC) \(|\Gamma_m(X_1)|\leq (m!)^{1+\gamma}/\Delta^{m-2},\;\;m=3,\;4,\dots \) (\(\Gamma_m(X)\) is an \(m\)th order cumulant of a r.v. \(X\)), for large-deviation probabilities and the case of i.i.d. random variables \(X_i\) with \(EX_1=0,\;DX_1=1.\) The tools used here are based on the cumulant's theory and just like that of \textit{L. Saulis} and \textit{V. Statulevičius} [``Limit theorems for large deviations'' (1989; Zbl 0714.60018)]. Results obtained here may be called the generalization of \textit{N. N. Amosova} [Theory Probab. Math. Stat. 32, 1-7 (1986); translation from Teor. Veroyatn. Mat. Stat. 32, 3-9 (1985; Zbl 0627.60033)], because SC is a slightly weakened Cramer's condition.