{
  "id": "1907.10379",
  "version": "v1",
  "published": "2019-07-24T12:13:36.000Z",
  "updated": "2019-07-24T12:13:36.000Z",
  "title": "Asymptotic Independence ex machina -- Extreme Value Theory for the Diagonal BEKK-ARCH(1) Model",
  "authors": [
    "Sebastian Mentemeier",
    "Olivier Wintenberger"
  ],
  "categories": [
    "math.PR"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-24T12:13:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "extreme value theory",
      "asymptotic independence",
      "diagonal bekk-arch",
      "multivariate regular variation properties",
      "multivariate stationary processes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}