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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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