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 propertiesof 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.


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