Weak convergence of partial maxima processes in the $M_{1}$ topology

Abstract

It is known that for a sequence of independent and identically distributed random variables $(X_{n})$ the regular variation condition is equivalent to weak convergence of partial maxima $M_{n}= \max\{X_{1}, \ldots, X_{n}\}$, appropriately scaled. A functional version of this is known to be true as well, the limit process being an extremal process, and the convergence takes place in the space of c\`{a}dl\`{a}g functions endowed with the Skorohod $J_{1}$ topology. We first show that weak convergence of partial maxima $M_{n}$ holds also for a class of weakly dependent sequences under the joint regular variation condition. Then using this result we obtain a corresponding functional version for the processes of partial maxima $M_{n}(t) = \bigvee_{i=1}^{\lfloor nt \rfloor}X_{i},\,t \in [0,1]$, but with respect to the Skorohod $M_{1}$ topology, which is weaker than the more usual $J_{1}$ topology. We also show that the $M_{1}$ convergence generally can not be replaced by the $J_{1}$ convergence. Applications of our main results to moving maxima, squared GARCH and ARMAX processes are also given.