A Note on "Quasi-Maximum-Likelihood Estimation in Conditionally Heteroscedastic Time Series: A Stochastic Recurrence Equations Approach"
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| Format: | Preprint |
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2025
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| _version_ | 1866916993635975168 |
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| author | Krabbe, Frederik |
| author_facet | Krabbe, Frederik |
| contents | 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 $γ> 1$. 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). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_05716 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Note on "Quasi-Maximum-Likelihood Estimation in Conditionally Heteroscedastic Time Series: A Stochastic Recurrence Equations Approach" Krabbe, Frederik Probability Statistics Theory 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 $γ> 1$. 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). |
| title | A Note on "Quasi-Maximum-Likelihood Estimation in Conditionally Heteroscedastic Time Series: A Stochastic Recurrence Equations Approach" |
| topic | Probability Statistics Theory |
| url | https://arxiv.org/abs/2510.05716 |