One technique of nonlinear normalization of random variables (Q1175856)

In connection with \textit{D. A. Darling}'s result [Trans. Am. Math. Soc. 73, 95--107 (1952; Zbl 0047.37502)] on the influence of the maximum term in the additions of i.r.v.s, the author considers the question of finding better nonlinear normalizations for sequences \(\{Z_n\}\) of sums (or maxima) of i.i.d. r.v.s. He investigates convergences of type \(a_n\varphi(Z_n-b_n)\underset n{\overset d{\rightarrow}} Z\), where \(b_n\) and \(a_n\) are real numbers, \(a_n>0\), \(a_n/a_{n+1}\to 1\), \(\varphi\) is continuous and strictly increasing.