Аннотация:A sequence of accompanying laws is suggested in the limit theoremof B. V. Gnedenko for maximums of independent random variables belonging tomaximum domain of attraction of the Gumbel distribution. It is shown that thissequence gives an exponential power rate of convergence whereas the Gumbeldistribution gives only a logarithmic rate. As examples, classes of Weibulland log-Weibull type distributions are considered in details. A scale for theGumbel maximum domain of attraction is suggested as a continuation of theconsidered two classes.