Weak Convergence of Partial Maxima Processes in the M1 Topology

It is known that for a sequence of independent and identically distributed random variables (Xn) the regular variation condition is equivalent to weak convergence of partial maximaMn = max{X1, . . . , Xn}, 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 J1 topology. We first show that weak convergence of partial maxima Mn 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 Mn(t) = ∨bntc i=1 Xi, t ∈ [0, 1], but with respect to the Skorohod M1 topology, which is weaker than the more usual J1 topology. We also show that the M1 convergence generally can not be replaced by the J1 convergence. Applications of our main results to moving maxima, squared GARCH and ARMAX processes are also given.