Asymptotic Independence ex machina - Extreme Value Theory for the Diagonal Stochastic Recurrence Equation

Sebastian Mentemeier , Olivier Wintenberger (1)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

We consider multivariate stationary processes $(\boldsymbol{X}_t)$ satisfying a stochastic recurrence equation of the form $$ \boldsymbol{X}_t= \boldsymbol{m}M_t \boldsymbol{X}_{t-1} + \boldsymbol{Q}_t,$$ where $(M_t)$ and $(\boldsymbol{Q}_t)$ are iid random variables and random vectors, respectively, and $\boldsymbol{m}=\mathrm{diag}(m_1, \dots, m_d)$ is a deterministic diagonal matrix. 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 if and only if $m_i \neq m_j$; 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 multivariate autoregressive conditional heteroskedasticity (ARCH) processes.

Mots clés: AMS 2010 subject classifications: 60G70; 60G10 Keywords: Stochastic recurrence equations; multivariate ARCH; multivariate regular variation; non-standard regular variation

Domaines: Statistiques [math.ST]

Sebastian Mentemeier, Olivier Wintenberger. Asymptotic Independence ex machina - Extreme Value Theory for the Diagonal Stochastic Recurrence Equation. 2020. ⟨hal-02591409v1⟩