Krizmanić, Danijel Functional weak convergence of partial maxima processes. (English) Zbl 1333.60062 Extremes 19, No. 1, 7-23 (2016). Summary: 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 Skorokhod \(J_{1}\) 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. Cited in 5 Documents MSC: 60F17 Functional limit theorems; invariance principles 60G52 Stable stochastic processes 60G70 Extreme value theory; extremal stochastic processes Keywords:extremal index; functional limit theorem; regular variation; Skorokhod \(J_{1}\) topology; strong mixing; weak convergence PDFBibTeX XMLCite \textit{D. Krizmanić}, Extremes 19, No. 1, 7--23 (2016; Zbl 1333.60062) Full Text: DOI arXiv References: [1] Adler, R.J.: Weak convergence results for extremal processes generated by dependent random variables. Ann. Probab. 6, 660-667 (1978) · Zbl 0377.60027 [2] Basrak, B., Krizmanić, D., Segers, J.: A functional limit theorem for partial sums of dependent random variables with infinite variance. Ann. Probab. 40, 2008-2033 (2012) · Zbl 1295.60041 [3] Basrak, B., Segers, J.: Regularly varying multivariate time series. Stochastic Process. Appl. 119, 1055-1080 (2009) · Zbl 1161.60319 [4] Breiman, L.: On some limit theorems similar to arc-sin law. Theory Probab. 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