Moments and Cumulants

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.

$$g(\lambda )=\sum _{k=-\infty }^\infty \mathrm {Cov}\,(X_0,X_k)e^{-ik\lambda }.$$

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

$$g(\lambda _2,\ldots ,\lambda _p)=\sum _{k_2=-\infty }^\infty \!\!\cdots \!\!\sum _{k_p=-\infty }^\infty \kappa (X_0,X_{k_2},\ldots , X_{k_p}) e^{-i(k_2\lambda _2+\cdots +k_p\lambda _p)}.$$