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A Multivariate Functional Limit Theorem in Weak $$M_{1}$$ M 1 Topology

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  • Bojan Basrak

    (University of Zagreb)

  • Danijel Krizmanić

    (University of Rijeka)

Abstract

We show a new functional limit theorem for weakly dependent regularly varying sequences of random vectors. As it turns out, the convergence takes place in the space of $$\mathbb R ^{d}$$ R d valued càdlàg functions endowed with the so-called weak $$M_{1}$$ M 1 topology. The theory is illustrated on two examples. In particular, we demonstrate why such an extension of Skorohod’s $$M_1$$ M 1 topology is actually necessary for the limit theorem to hold.

Suggested Citation

  • Bojan Basrak & Danijel Krizmanić, 2015. "A Multivariate Functional Limit Theorem in Weak $$M_{1}$$ M 1 Topology," Journal of Theoretical Probability, Springer, vol. 28(1), pages 119-136, March.
    • Handle: RePEc:spr:jotpro:v:28:y:2015:i:1:d:10.1007_s10959-013-0510-3
    DOI: 10.1007/s10959-013-0510-3
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    References listed on IDEAS

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    1. Basrak, Bojan & Davis, Richard A. & Mikosch, Thomas, 2002. "Regular variation of GARCH processes," Stochastic Processes and their Applications, Elsevier, vol. 99(1), pages 95-115, May.
    2. Hafner, Christian M. & Preminger, Arie, 2009. "Asymptotic Theory For A Factor Garch Model," Econometric Theory, Cambridge University Press, vol. 25(2), pages 336-363, April.
    3. Basrak, Bojan & Segers, Johan, 2009. "Regularly varying multivariate time series," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1055-1080, April.
    4. Denker, Manfred & Jakubowski, Adam, 1989. "Stable limit distributions for strongly mixing sequences," Statistics & Probability Letters, Elsevier, vol. 8(5), pages 477-483, October.
    5. Jakubowski, Adam & Kobus, Maria, 1989. "[alpha]-Stable limit theorems for sums of dependent random vectors," Journal of Multivariate Analysis, Elsevier, vol. 29(2), pages 219-251, May.
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