A POINT-PROCESS APPROACH TO ALMOST-SURE BEHAVIOR OF RECORD VALUES ANDORDER-STATISTICS

Point-process and other techniques are used to make a comprehensive in vestigation of the almost-sure behaviour of partial maxima M(n)((r)) ( the rth largest among a sample of n i.i.d. random variables), partial record values R(n)((r)), and differences and quotients involving them. In particular, we obtain characterizations of such asymptotic propert ies as lim sup(n-->infinity) (M(n)((r)) - M(n)((r+s))) less than or eq ual to c a.s. for some finite constant c, or R(n+s)((r)) - R(n)((r))-- >c a.s. for some constant c in [0,infinity], which tell us, in various ways, how quickly the sequences increase. These characterizations tak e the form of integral conditions on the tail of F, which furthermore characterize such properties as stability and relative stability of th e sequence of maxima. We also develop their relation to the large-samp le behaviour of trimmed sums, and discuss some statistical application s.