, Volume 17, Issue 3, pp 447–465 | Cite as

Weak convergence of partial maxima processes in the M 1 topology



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 càdlàg 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.


Functional limit theorem Regular variation Extremal process M1 topology Weak convergence 

AMS 2000 Subject Classifications

60F05 60F17 60G52 60G70 


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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.University of RijekaRijekaCroatia

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