Abstract
This chapter is devoted to moment methods. The use of moments relies on their importance in deriving asymptotic of several estimators, based on moments and limit distributions. Cumulants are linked with spectral or multispectral estimation which are main tools of time series analysis.
Such functions do not characterize the dependence of non-linear processes; indeed we have already examples of orthogonal and non-independent sequences. This motivates the introduction of higher order characteristics. A multispectral density is defined over \(\mathbb {C}^{p-1}\) by
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Notes
- 1.
This holds if there exists \(\alpha >0\) with \(\mathbb {E}e^{\alpha |U|}<\infty \).
- 2.
These formulae are proved for example in Rosenblatt (1985), pp. 33–34.
- 3.
The function \(s\mapsto \log (1+s)\) is analytic for \(|t|<1\), and the determination of the logarithm is not a problem in the domain \(]-\frac{1}{2},\frac{1}{2}[\) of \(\mathbb {C}\).
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Doukhan, P. (2018). Moments and Cumulants. In: Stochastic Models for Time Series. Mathématiques et Applications, vol 80. Springer, Cham. https://doi.org/10.1007/978-3-319-76938-7_12
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DOI: https://doi.org/10.1007/978-3-319-76938-7_12
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Online ISBN: 978-3-319-76938-7
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