On the Number of Records Near the Maximum

summary Recent work has considered properties of the number of observationsXj, independently drawn from a discrete law, which equal the sample maximumX(n)The natural analogue for continuous laws is the numberKn(a) of observations in the interval (X(n)–a,X(n)], wherea> 0. This paper derives general expressions for the law, first moment, and probability generating function ofKn(a), mentioning examples where evaluations can be given. It seeks limit laws forn→ and finds a central limit result whenais fixed and the population law has a finite right extremity. Whenever the population law is attracted to an extremal law, a limit theorem can be found by lettingadepend onnin an appropriate manner; thus the limit law is geometric when the extremal law is the Gumbel type. With these results, the paper obtains limit laws for ‘top end’ spacingsX(n)‐X(n‐j)withjfixed.