Maximum of continuous versions of Poisson and negative binomial-type distributions

摘要

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→ Y, nondegenerate, as n→∞ if and only if F(x)/F(x-) → 1 as x→∞, where F(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(x) = cx~β exp(αx)x~(-x){1+o(1)} and F(x) = dxc exp(-ax){1+o(1)} as x→∞. 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→ Y as n → ∞ for some an and bn, where Y is a Gumbel random variable with distribution function exp{- exp(-y)}, -∞ < y < ∞.