The Logistic Distribution as a Limit Law for Random Sums and Statistics Constructed from Samples with Random Sizes

Abstract

In the present paper, based on the representation of the logistic distribution as a normal scale mixture obtained by L. Stefanski in 1990, it is demonstrated that the logistic distribution can be limiting for sums of a random number of random variables and other statistics that admit (at least asymptotically) additive representation and are constructed from samples with random sizes. These results complement a theorem proved by B. V. and D. B. Gnedenko in 1982 that established convergence of the distributions of extreme order statistics in samples with geometrically distributed random sizes to the logistic distribution. Hence, along with the normal law, this distribution can be used as an asymptotic approximation of the distributions of observations that can be assumed to have an additive structure, for example, random-walk-type time series. An approach is presented for the definition of the new asymmetric generalization of the logistic distribution as a special normal variance–mean mixture.

Keywords

• logistic distribution
• normal scale mixture
• Kolmogorov distribution
• limit theorem
• random sum
• normal variance-mean mixture