Asymptotic results for sample autocovariance functions and extremes of integrated generalized Ornstein–Uhlenbeck processes

Vicky Fasen

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We consider a positive stationary generalized Ornstein–Uhlenbeck process $$V_t=e^{−ξ_t} \left( ∫_0^te^{ ξ_{s−}}dη_s+V_0 \right)  \mathrm{for}\quad t≥0,$$ and the increments of the integrated generalized Ornstein–Uhlenbeck process $I_{k}=\int_{k-1}^{k}\sqrt{V_{t-}}\,\mathrm{d}L_{t}$, $k∈ℕ$, where $(ξ_t, η_t, L_t)_{t≥0}$ 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 ARCH(1) and GARCH(1, 1) processes. In this paper we investigate the asymptotic behavior of extremes and the sample autocovariance function of $(V_t)_{t≥0}$ and $(I_k)_{k∈ℕ}$. Furthermore, we present a central limit result for $(I_k)_{k∈ℕ}$. Regular variation and point process convergence play a crucial role in establishing the statistics of $(V_t)_{t≥0}$ and $(I_k)_{k∈ℕ}$. The theory can be applied to the COGARCH$(1, 1)$ and the Nelson diffusion model.

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Bernoulli, Volume 16, Number 1 (2010), 51-79.

First available in Project Euclid: 12 February 2010

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continuous-time GARCH process extreme value theory generalized Ornstein–Uhlenbeck process integrated generalized Ornstein–Uhlenbeck process mixing point process regular variation sample autocovariance function stochastic recurrence equation


Fasen, Vicky. Asymptotic results for sample autocovariance functions and extremes of integrated generalized Ornstein–Uhlenbeck processes. Bernoulli 16 (2010), no. 1, 51--79. doi:10.3150/08-BEJ174.

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