Rates of convergence in certain limit theorem for extreme values

Abstract

Let Xn 1*, ..., Xn n* be independent random variables with the common negative binomial distribution with parameters r > 0 and 1 / n, where r is not necessarily an integer. We determine the limiting distribution of the random variable Mn* = max {Xn 1*, ..., Xn n*} as n → ∞, corresponding normalizing constants and the rate of convergence. For an integer r the connection with certain waiting time problems is indicated. © 2009 Elsevier Inc. All rights reserved.