Discretization of distributions in the maximum domain of attraction

In this paper we discuss the discretization of distributions belonging to some max-domain of attraction. Given a random variable X its discretization is defined as the minimal integer not less than X. Our first interest is on distributions that preserve the max-domain property after discretization. Secondly, we characterize the distributions which are regarded as the discretization of the distribution in the Gumbel max-domain of attraction. Lastly the correspondence of distribution in Gumbel max-domain of attraction is investigated.