Abstract
Some bases for the theory of time series are given below. The chapter deals with the widely used assumption of stationarity which yields a simpler theory for time series. This concept is widely considered in Rosenblatt (Stationary processes and random fields, Birkhäuser, Boston, 1985) and in Brockwell and Davis (Time series: theory and methods, 2nd edn., Springer series in statistics, Springer, New York, 1991). The latter reference is more involved with linear time series.
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Notes
- 1.
Nile data may be found on: https:datamarket.com/dataset22w8mean-annual-nile-flow-1871-1970.
- 2.
Use a triangular scheme, by successive extraction of convergent subsequences. Choose a denumerable and dense sequence \((\lambda _k)_k\) in \([-\pi ,\pi ]\).
Here \(\phi _{k+1}(n)\) will denote a subsequence of \(\phi _k(n)\) such that \(G_{\phi _{k+1}(n)}(\lambda _{k+1})\) converges as \(n\rightarrow \infty \).
Setting \(G_{\phi (n)}=G_{\phi _{n}}(n)\) allows to end the proof.
- 3.
Recall that monotonic functions admit limits on the left and on the right at each point, the non-empty open intervals \((f(x-), f(x+))\) are disjoint in \(\mathbb {R}\). Choose a rational number in each of them to conclude.
- 4.
This process is the centred Gaussian process indexed on \(\mathbb {R}^+\) with the covariance \(\mathbb {E}W(s)W(t)=s\wedge t\).
- 5.
Landau notations:
\(v_n=o(u_n)\) if \(\lim _n(v_n/u_n)=0\) in case \(u_n\ne 0\) for all n,
\(v_n=\mathcal{O}(u_n)\) if there exists \(C>0\) such that \(|v_n|\le C|u_n|\) for all n.
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Doukhan, P. (2018). Stationarity. In: Stochastic Models for Time Series. Mathématiques et Applications, vol 80. Springer, Cham. https://doi.org/10.1007/978-3-319-76938-7_4
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DOI: https://doi.org/10.1007/978-3-319-76938-7_4
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