Limit laws for mixtures with applications to asymptotic theory of extremes

For a sequence X 1 , X 2 , . . . of random variables, consider the events Aj(x)={Xj>x}, j = 1, 2, ..., where x is an arbitrary real number. Putting v,(x) for the number of Al(x),A2(x),...,A,(x) which occur, the event {v,(x)=0} reduces to {Z,<x}, where Z,=max{X1,X 2 . . . . ,X,}. Here n can be a given integer or a random variable itself. This research has, in fact, started with the aim of unifying techniques for proving limit laws for the extremes when (i) the X's are independent and n is a random variable, independently distributed of the X's and when (ii) the X's are from an infinite sequence of exchangeable random variables and n is a fixed integer. The common property of these two cases is that the distribution of v, (x) can be written in the form