## A note on calculating autocovariances of long-memory processes.(English)Zbl 1062.62164

This paper deals with a splitting method for calculating the autocovariances of fractional integrated processes ARFIMA and generalized integrated processes GARMA. Autocovariances of the ARFIMA$$(0,d,0)$$ process and of the GARMA(0,0) process are defined as $\gamma^*(k)=\sigma^2\Gamma(k+1)\Gamma(1-2d)(\Gamma(k+1-d)\Gamma(1-d)\Gamma(d))^{-1},$
$\gamma^*=(\sigma^2/2\sqrt{\pi})\Gamma(1-2\lambda)(2\sin\nu)^{1/2-2\lambda}[P^{2\lambda-1/2}_{k-1/2}(\eta)+ (-1)^{k}P^{2\lambda-1/2}_{k-1/2}(-\eta)],$ where $$\nu=\cos^{-1}(\eta)$$ and $$P_{a}^{b}(c)$$ is the associate Legendre function of the first kind, $$\Gamma(x)$$ is the Gamma function, and $$\sigma^2$$ is the variance of the noise. The authors consider a group of ARFIMA and GARMA models with $$p\neq0$$ and/or $$q\neq0$$. The long memory models are studied numerically, considering parameters $$d$$ and $$\lambda$$ ranging in $$(0,1/2)$$. In the case of $$d$$ and $$\lambda\in(-1/2,0)$$ results are similar. The considered methods generate close theoretical values but splitting is faster and more precise in the case of GARMA$$(p,q)$$ processes.

### MSC:

 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 65C60 Computational problems in statistics (MSC2010)
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### References:

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