MAXIMUM OF CONTINUOUS VERSIONS OF POISSON AND NEGATIVE BINOMIAL-TYPE DISTRIBUTIONS

Let M-n be the maximum of a random sample from F(x) on R. It is known that there exists g(n) : R -> R such that g(n)(M-n) (L) under right arrow Y, nondegenerate, as n -> infinity if and only if (F) over bar (x)/(F) over bar (x-) -> 1 as x -> infinity, where (F) over bar (x) = 1-F(x). This condition holds for F continuous but fails, for example, for F Poisson and negative binomial. We consider the classes of distributions (F) over bar (x) = cx(beta) exp(alpha x) x(-x) {1+ o(1)} and (F) over bar (x) = dx(c) exp(-alpha x){1+ o(1)} as x -> infinity. These classes include the Poisson and negative binomial distributions for x an integer but not for general x. We show that (M-n - a(n))/b(n) (L) under righht arrow Y as n -> infinity for some a(n) and b(n), where Y is a Gumbel random variable with distribution function exp{- exp(-y)}, - infinity < y < infinity.