Let \(c_ j(F)\) denote the \(j\)-th cumulant of the distribution function \(F\). The main result of the paper can be formulated as follows. Let \(\{F_ n\}\) be a sequence of distribution functions and let \(F\) be a distribution function which is specified by its higher-order cumulants (this notion is defined in the appropriate way, similar as in the case of moments). If there exists a positive integer \(J\) such that, as \(n\to\infty\), \(c_ j(F_ n)\to c_ j(F)\), \(j=1,2\) and \(j\geq J\), then \(F_ n\) converges weakly to \(F\). It is an open problem to characterize the family of distributions which are specified by their higher-order cumulants. For previous works on this topic see [\textit{V. A. Malyshev}, Soviet Math., Dokl. 16(1975), 1141-1145; translation from Akad. Nauk SSSR, Dokl. 224, No. 1, 35-38 (1975)], [\textit{J. T. Cox} and \textit{G. R. Grimmett}, J. Stat. Phys. 25, 237-251 (1981; Zbl 0512.60094)], and [\textit{S. Janson}, Ann. Probab. 16, No. 1, 305-312 (1988; Zbl 0639.60029)].