Rates of convergence in certain limit theorem for extreme values

Let X n 1 * , … , X n 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 M n * = max { X n 1 * , … , X n n * } as n → ∞ , corresponding normalizing constants and the rate of convergence. For an integer r the connection with certain waiting time problems is indicated.