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Identification of persistent cycles in non-Gaussian long-memory time series. (English) Zbl 1198.62089
The author deals with the autoregressive model $\phi(B)y_t=\varepsilon_t,$ where $y_t$ is the $t$-th observation on the dependent variable, $y_t=0$ for $t<0$, $\phi(B)=1-\phi_1B-\cdots-\phi_pB^p$ is the characteristic polynomial, $B$ is the backward shift operator, i.e., $By_t=y_{t-1}$, and the disturbance process $\varepsilon_t$ is given by $\varepsilon_t=\sum_{j\leq t}b_{t-j}u_j$, where $\{u_j\}$ is a sequence of i.i.d. $(0,1)$ random variables, $\{b_j\}$ is a sequence which decays hyperbolically, i.e., $b_j=j^{H_3/2}L_1(j)$, $0<H<1$, $\sum_{j=0}^{\infty}b_j^2<\infty$, and $L_1(j)$ is a slowly varying function, bounded on every finite interval. The unknown parameter $\Phi=(\phi_1,\dots,\phi_p)$ is estimated by the least squares estimate (LSE): $$\hat{\Phi}=\left( \sum_{k=1}^n {\bold y}_{k-1}{\bold y}'_{k-1} \right)^{-1}\sum_{k=1}^n {\bold y}_{k-1}{y}_{k},$$ where ${\bold y}_{k1}=(y_k,\dots,y_{k-p+1})'$. The least squares error satisfies $$\hat{\Phi}-\Phi=\left( \sum_{k=1}^n {\bold y}_{k-1}{\bold y}'_{k-1} \right)^{-1}\sum_{k=1}^n {\bold y}_{k-1}{\varepsilon}_{k}.$$ The asymptotic distribution is derived for the least squares estimates (LSE) only in the case when the characteristic polynomial $\phi(z)$ is unstable with complex-conjugate unit roots, i.e., an appropriate non-stationary model to identify persistent cycles in non-Gaussian long-memory time series. The behaviour of the LSE when $\phi(z)$ has stable roots, roots equal to -1 and 1, and explosive roots remains an open problem. In order to describe the limiting distribution of the LSE, two limit theorems involving long-memory processes are proved. The first theorem gives the limiting distribution of the weighted sum, $\sum_{k=1}^nc_{n,k}\varepsilon_k$, where $\varepsilon_k=\sum_{j\leq k}b_{k-j}u_j$ is a non-Gaussian long-memory moving-average process and $\{c_{n,k},1\leq k\leq n\}$ is a given sequence of weights. The second theorem is a functional central limit theorem for the sine and cosine Fourier transforms $\sum_{k=1}^{[nt]}\sin(k\theta)\varepsilon_k$, $\sum_{k=1}^{[nt]}\cos(k\theta)\varepsilon_k$, where $\theta\in (0,\pi)$ and $t\in[0,1]$.

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. (statistics) 62E20 Asymptotic distribution theory in statistics 60F17 Functional limit theorems; invariance principles
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