On conditions of convergence of the distributions of extremal order statistics to the Weibull distribution

Some product representations are obtained for random variables with the Weibull distribution by stable random variables. These results are used to describe the conditions providing convergence of the distributions of linearly transformed minimum order statistics in samples with random sizes to the Weibull distribution. The presented results broaden traditional conceptions concerning conditions of convergence of extremal order statistics to the Weibull distribution and give additional theoretical explanation for high adequacy of the Weibull distribution in lifetime data analysis, in particular, in reliability theory.