The Annals of Applied Probability

A characterization of multivariate regular variation

Bojan Basrak, Richard A. Davis, and Thomas Mikosch

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We establish the equivalence between the multivariate regular variation of a random vector and the univariate regular variation of all linear combinations of the components of such a vector. According to a classical result of Kesten [Acta Math. 131 (1973) 207-248], this result implies that stationary solutions to multivariate linear stochastic recurrence equations are regularly varying. Since GARCH processes can be embedded in such recurrence equations their finite-dimensional distributions are regularly varying.

Article information

Ann. Appl. Probab., Volume 12, Number 3 (2002), 908-920.

First available in Project Euclid: 12 September 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E05: Distributions: general theory
Secondary: 60G10: Stationary processes 60G55: Point processes 60G70: Extreme value theory; extremal processes 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62P05: Applications to actuarial sciences and financial mathematics

Point process vague convergence multivariate regular variation heavy tailed distribution stochastic recurrence equation GARCH process


Basrak, Bojan; Davis, Richard A.; Mikosch, Thomas. A characterization of multivariate regular variation. Ann. Appl. Probab. 12 (2002), no. 3, 908--920. doi:10.1214/aoap/1031863174.

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