engleski

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 convergence takes place in the space of $\mathbb{;R};^{;2};$--valued c\`{;a};dl\`{;a};g functions on $[0, 1]$, with the Skorohod weak $M_{;1};$ topology, and the limiting process consists of an $\alpha$--stable Levy process and an extremal process. We also describe the dependence between these two components of the limit, and show that the weak $M_{;1};$ topology in general can not be replaced by the standard $M_{;1};$ topology.

Functional convergence ; Regular variation ; Partial sum process ; Partial maxima process

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