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

Abstract : 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.
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Cited literature [21 references]

https://hal.archives-ouvertes.fr/hal-02591409
Contributor : Olivier Wintenberger <>
Submitted on : Friday, May 15, 2020 - 3:11:30 PM
Last modification on : Saturday, April 3, 2021 - 3:29:37 AM

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• HAL Id : hal-02591409, version 1
• ARXIV : 1907.10379

### Citation

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

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