Functional weak convergence of partial maxima processes

For a strictly stationary sequence of nonnegative regularly varying random variables $(X_{n})$ we study functional weak convergence of partial maxima processes $M_{n}(t) = \bigvee_{i=1}^{\lfloor nt \rfloor}X_{i},\,t \in [0,1]$ in the space $D[0,1]$ with the Skorohod $J_{1}$ topology. Under the strong mixing condition, we give sufficient conditions for such convergence when clustering of large values do not occur. We apply this result to stochastic volatility processes. Further we give conditions under which the regular variation property is a necessary condition for $J_{1}$ and $M_{1}$ functional convergences in the case of weak dependence. We also prove that strong mixing implies the so-called Condition $\mathcal{A}(a_{n})$ with the time component.

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