On the number of records near the maximum

Recent work has considered properties of the number of observations X-j, independently drawn from a discrete law, which equal the sample maximum X-(n). The natural analogue for continuous laws is the number K-n(a) of observations in the interval (X-(n)-a, X-(n)], where a > 0. This paper derives general expressions for the law, first moment, and probability generating function of K-n(a), mentioning examples where evaluations can be given. It seeks limit laws for n --> infinity, and finds a central Limit result when a is 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 letting a depend on n in 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' spacings X-(n)-Xn-j) with j fixed.