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.
"A characterization of multivariate regular variation." Ann. Appl. Probab. 12 (3) 908 - 920, August 2002. https://doi.org/10.1214/aoap/1031863174