## 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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### References:

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