Weak Convergence to Extremal Processes

Abstract

$\{X_n, n\geqq 1\}$ are i.i.d. rv's with df $F$. Set $M_n = \max\{X_1, \cdots, X_n\}$. As a basic assumption, suppose normalizing constants $a_n > 0, b_n, n \geqq 1$ exist such that $\lim_{n\rightarrow\infty} P\lbrack M_n \leqq a_n x + b_n \rbrack = G(x)$, nondegenerate. Define the random function $Y_n(t) = (M_{\lbrack nt \rbrack} - b_n)/a_n$. By considering weak convergence of underlying two dimensional point processes, an alternate proof of the original Lamperti result that $Y_n \Rightarrow Y$ is given where $Y$ is an extremal-$G$ process. From the convergence of the point processes, other weak convergence results are shown. Let $x(t)$ be nondecreasing and $Nx(I)$ be the number of times $x$ jumps in time interval $I$. Then $Y_n^{-1} \Rightarrow Y^{-1}, NY_n \Rightarrow NY, NY_n^{-1} \Rightarrow NY^{-1}$. From these convergences emerge a variety of limit results for record values, record value times and inter-record times.