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Estimation of limiting conditional distributions for the heavy tailed long memory stochastic volatility process. (English) Zbl 1273.60028
Consider a stochastic volatility process defined as $Y_{i}=\sigma\left( X_{i}\right) Z_{i},\mathstrut i\in\mathbb{Z}\text{,}$ where $$\sigma$$ is some (possibly unknown) positive function, $$\left\{ Z_{j},j\in\mathbb{Z}\right\}$$ is an i.i.d. sequence and $$\left\{ X_{j} ,j\in\mathbb{Z}\right\}$$ is a stationary Gaussian process with mean zero, unit variance, and independent from the process $$\left\{ X_{j}\right\}$$. The authors study certain extremal properties of the finite dimensional joint distributions of the process $$\left\{ Y_{j}\right\}$$ when $$Z_{1}$$ is heavy tailed and the Gaussian process $$\left\{ X_{j}\right\}$$ possibly has long memory.
For fixed positive integers $$h<m$$ and $$h^{\prime}\geqq0$$, Borel sets $$A\subset\mathbb{R}^{h}$$ and $$B\subset\mathbb{R}^{h^{\prime}+1}$$, they are interested in the limits $\rho\left( A,B,m\right) =\lim\limits_{t\rightarrow\infty}\mathbb{P}\left( \left( Y_{m},\dots,Y_{m+h^{\prime}}\right) \in B\mid\left( Y_{1} ,\dots,Y_{h}\right) \in tA\right) \text{.}$
The general aim of this paper is to investigate the existence of the limiting conditional distributions $$\rho\left( A,B,m\right)$$ and their statistical estimation. The asymptotic properties of estimators are studied.

##### MSC:
 60F05 Central limit and other weak theorems 60G70 Extreme value theory; extremal stochastic processes
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