Record Values and Maxima

Abstract

$\{X_n, n \geqq 1\}$ are $\operatorname{i.i.d.}$ random variables with continuous $\operatorname{df} F(x). X_j$ is a record value of this sequence if $X_j > \max \{X_1,\cdots, X_{j-1}\}$. We compare the behavior of the sequence of record values $\{X_{L_n}\}$ with that of the sample maxima $\{M_n\} = \{\max (X_1,\cdots, X_n)\}$. Conditions for the relative stability ($\operatorname{a.s.}$ and $\operatorname{i.p.}$) of $\{X_{L_n}\}$ are given and in each case these conditions imply the relative stability of $\{M_n\}$. In particular regular variation of $R(x) \equiv - \log (1 - F(x))$ is an easily verified condition which insures $\operatorname{a.s.}$ stability of $\{X_{L_n}\}, \{M_n\}$ and $\{\sum^n_{j=1} M_j\}$. Concerning limit laws, $X_{L_n}$ may converge in distribution without $\{M_n\}$ having a limit distribution and vice versa. Suitable differentiability conditions on $F(x)$ insure that both sequences have a limit distribution.