# Asymptotic Independence ex machina - Extreme Value Theory for the Diagonal SRE Model

Abstract : 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. % It is known that under suitable assumptions the marginals $X_{t,i}$ of $\boldsymbol{X}_t$ are regularly varying. We obtain a full characterization of the Vector Scaling Regular Variation properties of $(\boldsymbol{X}_t)$, proving that some coordinates $X_{t,i}$ and $X_{t,j}$ are asymptotically independent even though all coordinates rely on the same random input $(M_t)$. We prove the asynchrony of extreme clusters among marginals with different tail indices. Our results are applied to some multivariate autoregressive conditional heteroskedastic (BEKK-ARCH and CCC-GARCH) processes and to log-returns. Angular measure inference shows evidences of asymptotic independence among marginals of diagonal SRE with different tail indices.
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Journal articles
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https://hal.archives-ouvertes.fr/hal-02591409
Contributor : Olivier Wintenberger Connect in order to contact the contributor
Submitted on : Tuesday, January 4, 2022 - 5:23:56 PM
Last modification on : Friday, January 7, 2022 - 3:40:12 AM

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### Identifiers

• HAL Id : hal-02591409, version 3
• ARXIV : 1907.10379

### Citation

Sebastian Mentemeier, Olivier Wintenberger. Asymptotic Independence ex machina - Extreme Value Theory for the Diagonal SRE Model. Journal of Time Series Analysis, Wiley-Blackwell, In press. ⟨hal-02591409v3⟩

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