On accompanying measures and asymptotic expansions in the B. V. Gnedenko limit theorem

We propose a sequence of accompanying laws in the B. V. Gnedenko limit theorem for maxima of independent random variables with distributions lying in the Gumbel max domain of attraction. We show that this sequence provides a power-law convergence rate, whereas the Gumbel distribution provides only the logarithmic rate. As examples, we consider in detail the classes of Weibull and log-Weibull type distributions. For the entire Gumbel max domain of attraction, we propose a scale of classes of distributions that includes these two classes as a starting point.