{"ID":2859234,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.05716","arxiv_id":"2510.05716","title":"A Note on \"Quasi-Maximum-Likelihood Estimation in Conditionally Heteroscedastic Time Series: A Stochastic Recurrence Equations Approach\"","abstract":"Bougerol (1993) and Straumann and Mikosch (2006) gave conditions under which there exists a unique stationary and ergodic solution to the stochastic difference equation $Y_t \\overset{a.s.}{=} Φ_t (Y_{t-1}), t \\in \\mathbb{Z}$ where $(Φ_t)_{t \\in \\mathbb{Z}}$ is a sequence of stationary and ergodic random Lipschitz continuous functions from $(Y,|| \\cdot ||)$ to $(Y,|| \\cdot ||)$ where $(Y,|| \\cdot ||)$ is a complete subspace of a real or complex separable Banach space. In the case where $(Y,|| \\cdot ||)$ is a real or complex separable Banach space, Straumann and Mikosch (2006) also gave conditions under which any solution to the stochastic difference equation $\\hat{Y}_t \\overset{a.s.}{=} \\hatΦ_t (\\hat{Y}_{t-1}), t \\in \\mathbb{N}$ with $\\hat{Y}_0$ given where $(\\hatΦ_t)_{t \\in \\mathbb{N}}$ is only a sequence of random Lipschitz continuous functions from $(Y,|| \\cdot ||)$ to $(Y,|| \\cdot ||)$ satisfies $γ^t || \\hat{Y}_t - Y_t || \\overset{a.s.}{\\rightarrow} 0$ as $t \\rightarrow \\infty$ for some $γ\u003e 1$. In this note, we give slightly different conditions under which this continues to hold in the case where $(Y,|| \\cdot ||)$ is only a complete subspace of a real or complex separable Banach space by using close to identical arguments as Straumann and Mikosch (2006).","short_abstract":"Bougerol (1993) and Straumann and Mikosch (2006) gave conditions under which there exists a unique stationary and ergodic solution to the stochastic difference equation $Y_t \\overset{a.s.}{=} Φ_t (Y_{t-1}), t \\in \\mathbb{Z}$ where $(Φ_t)_{t \\in \\mathbb{Z}}$ is a sequence of stationary and ergodic random Lipschitz conti...","url_abs":"https://arxiv.org/abs/2510.05716","url_pdf":"https://arxiv.org/pdf/2510.05716v1","authors":"[\"Frederik Krabbe\"]","published":"2025-10-07T09:27:24Z","proceeding":"math.PR","tasks":"[\"math.PR\",\"math.ST\"]","methods":"[]","has_code":false}