Functional weak convergence of partial maxima processes

TL;DR

This paper establishes conditions for the weak convergence of partial maxima processes of stationary, regularly varying sequences in the Skorohod space, with applications to stochastic volatility models and dependence structures.

Contribution

It provides new sufficient conditions for functional weak convergence of maxima processes under strong mixing and explores the necessity of regular variation for such convergence.

Findings

Weak convergence of maxima processes under strong mixing.

Application to stochastic volatility processes.

Regular variation as a necessary condition for convergence.

Abstract

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.