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Asymptotic results for long memory LARCH sequences. (English) Zbl 1032.62078

Let \((y_n,\sigma_n)\) be a long memory LARCH sequence defined by \[ y_k= \sigma_k\varepsilon_k,\quad \sigma_k= a+\sum^\infty_{i=1} b_i y_{k-i}, \] where \(a\neq 0\), \(\{\varepsilon_i\}\) are i.i.d. random variables with \(E\varepsilon_0= 0\), \(E\varepsilon^2_0= 1\), \(E|\varepsilon_0|^p< \infty\) \((p> 4)\), and \(\sum_n b^2_n<(p- 1)/(3(6p)^3\|\varepsilon_0\|^2_p)\). Denote by \(\{W_\gamma(t), t\geq 0\}\) the fractional Brownian motion with parameter \(\gamma\) \((0<\gamma< 1)\). The authors prove that if \(f\) is a twice continuously differentiable function with \(|f''(x)|\leq C(|x|^\alpha+ 1)\) \((0< \alpha< (p- 4)^2/(2p))\), then \[ N^{-3/2- \beta} \sum^{[Nt]}_{n= 1} (f(\sigma_n)- Ef(\sigma_n))@>{\mathcal D}[0, 1]>>\gamma dW_{(3/2)- \beta}(t), \] and \[ N^{-3/2- \beta} \sum^{[Nt]}_{n= 1} (f(y_n)- Ef(y_n))@>{\mathcal D}[0, 1]>> \gamma_1 dW_{(3/2)- \beta}(t), \] where \(d\) is a positive constant and \(\gamma\), \(\gamma_1\) are some constants depending on \(E(\sigma_0f'(\sigma_0))\) and \(E(y_0 f'(y_0))\), respectively. Moreover, for some new LARCH type sequences associated with \((\sigma_n)\), they formulate a stronger version of the above result.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60K99 Special processes
60F17 Functional limit theorems; invariance principles
62E20 Asymptotic distribution theory in statistics
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