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math.ST updates on arXiv.org

What is Learnable in Valiant's Theory of the Learnable? 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Asymptotic results for sample autocovariance functions and extremes of integrated generalized Ornstein-Uhlenbeck processes
Vicky Fasen · 2010-02-23 · via math.ST updates on arXiv.org

We consider a positive stationary generalized Ornstein--Uhlenbeck process \[V_t=\mathrm{e}^{-ξ_t}\biggl(\int_0^t\mathrm{e}^{ξ_{s-}}\ ,\mathrm{d}η_s+V_0\biggr)\qquadfor t\geq0,\] and the increments of the integrated generalized Ornstein--Uhlenbeck process $I_k=\int_{k-1}^k\sqrt{V_{t-}} \mathrm{d}L_t$, $k\in\mathbb{N}$, where $(ξ_t,η_t,L_t)_{t\geq0}$ is a three-dimensional Lévy process independent of the starting random variable $V_0$. The genOU model is a continuous-time version of a stochastic recurrence equation. Hence, our models include, in particular, continuous-time versions of $\operatorname {ARCH}(1)$ and $\operatorname {GARCH}(1,1)$ processes. In this paper we investigate the asymptotic behavior of extremes and the sample autocovariance function of $(V_t)_{t\geq0}$ and $(I_k)_{k\in\mathbb{N}}$. Furthermore, we present a central limit result for $(I_k)_{k\in\mathbb{N}}$. Regular variation and point process convergence play a crucial role in establishing the statistics of $(V_t)_{t\geq0}$ and $(I_k)_{k\in\mathbb{N}}$. The theory can be applied to the $\operatorname {COGARCH}(1,1)$ and the Nelson diffusion model.

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