Rates of convergence in certain limit theorem for extreme values (Q1043928)

Consider an array of random variables \(X_{n1},\dots,X_{nn}\), \(n\in\mathbb{N}\), where in each row \(X_{n1},\dots,X_{nn}\) are independent with common negative binomial distribution with parameters \(r>0\) and \(p=1/n\), \(r\) not necessarily being an integer. It is shown that the limiting distribution of the maximum in each row, suitably standardized, is the Gumbel distribution \(\exp(-e^{-x})\), \(x\in\mathbb{R}\). The rate of pointwise convergence of the distribution functions is established as well. This extends earlier results by the author that were formulated for integer-valued \(r\).