We consider multivariate stationary processes $(\boldsymbol{ X}_t)$ satisfying a stochastic recurrence equation of the form $$ \boldsymbol{ X}_t= \mathbb{ M}_t \boldsymbol{ X}_{ t-1} + \boldsymbol{ Q}_t,$$ where $(\boldsymbol{ Q}_t)$ are iid random vectors and $$ \mathbb{ M}_t=\mathrm{ Diag}(b_1+c_1 M_t, \dots, b_d+c_d M_t) $$ are iid diagonal matrices and $(M_t)$ are iid random variables. We obtain a full characterization of the multivariate regular variation properties of $(\boldsymbol{ X}_t)$, proving that coordinates $X_{ t,i}$ and $X_{ t,j}$ are asymptotically independent even though all coordinates rely on the same random input $(M_t)$. We describe extremal properties of $(\boldsymbol{ X}_t)$ in the framework of vector scaling regular variation. Our results are applied to some multivariate autoregressive conditional heteroskedasticity (BEKK-ARCH and CCC-GARCH) processes.
Categories: math.PR