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On joint weak convergence of partial sum and maxima processes

For a strictly stationary sequence of random variables we derive functional convergence of the joint partial sum and partial maxima process under joint regular variation with index alpha \in (0, 2) and weak dependence conditions. The limiting process consists of an alpha-stable Lévy process and an extremal process. We also describe the dependence between these two components of the limit. The convergence takes place in the space of R^2- valued cadlag functions on [0, 1], with the Skorohod weak M1 topology. We further show that this topology in general can not be replaced by the stronger (standard) M1 topology.

Functional limit theorem ; regular variation ; weak M1 topology ; extremal process ; Lévy process

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