Theoretical Consideration of Asymptotic Distributions of Extremes for the Case of Log-Normal Distribution

Abstract

In this study, we investigate asymptotic distributions of extremes for the case when the initial distribution is log-normal. We show theoretically by means of logarithmic transformation that the second asymptotic distribution is derived for the largest value. This result seems to be in contradiction to Gumbel's statement that the first asymptotic distribution is derived for the largest value, since it is not possible for two different types of asymptotic distributions of the largest value to correspond to the same initial distribution. This apparent contradiction is explained by the finding that the resulting first and second asymptotic distributions coincide with each other due to some relation between their distribution parameters. This result is also demonstrated graphically using probability paper. We also examine the asymptotic distribution of the smallest value.