A generalization of the global limit theorems of R. P. Agnew (Q1103261)

\textit{R. P. Agnew} [Proc. Nat. Acad. Sci. USA. 40, 800-804 (1954; Zbl 0055.367)], studied the relationships between the weak convergence of distribution functions \(F_ n\) to a distribution function \(F_ 0\) and the convergence of \(\int_{R}\phi (| F_ n-F_ 0|)dx\) to 0 where \(\phi (t)=| t|^ r\). Sufficient and, separately, necessary conditions have been given, which are so-called global limit theorems. In this paper, these results are generalized. The sufficiency conclusions are sharp. Moreover, weak convergence of distribution functions is characterized in terms of their almost everywhere limiting behavior with respect to the Lebesgue measure on the line. Kruglov's techniques are employed to establish them.