##
**Stationary sequences and random fields.**
*(English)*
Zbl 0597.62095

Boston-Basel-Stuttgart: BirkhĂ¤user. 258 p. DM 110.00 (1985).

At a time when there is an abundance of publications on ”time series”, this book treats stationary sequences and fields in an original way by giving an important place to the foundations of numerous results.

This text is at the level of a postgraduate university course. It constitutes a useful reference on complementary probabilistic or statistical results for any person specialized in time series who wishes to learn more about the mathematical background. The techniques used for deriving limit theorems are ”non ordered” and the concept of martingale is not used. This allows the author to lead us gently into stationary random fields for which classical results in dimension one are extended in a natural way: central limit theorem, spectral estimation, Whittle’s parametric model estimation. Both time and frequency domain approaches are developed.

Let us outline two new topics discussed in this book. The C.L.T. for a field (or a sequence) is obtained with the following hypothesis: a strongly mixing field and the convergence to infinity of the \(L^{2+\delta}\) norm of the sums. Hence no conditions are imposed on the rate of decrease of the mixing coefficients and the techniques used, which are free of any ordering, are independent of the dimension k of the domain \(Z^ k\) on which the process is defined.

This C.L. result allows to prove subsequently the convergence to a normal distribution of the empirical covariances, of the smoothed spectrogram and of Whittle’s estimator for a field in \(L^ 8\) with summable cumulants of order 4. Let us note nevertheless an important difference between the cases \(k=1\) and \(k\geq 2\), a difference which is not pointed out by the author: because of edge effects the proposed estimators are \(n^{k/2}\) biased if \(k>1\) and the domain of observation is the hypercube of side n. To counter this, data on edges need to be tapered.

The second new topic which is approached concerns the study of non- Gaussian linear processes. First, a whole chapter is devoted to the estimation of the spectral density of the cumulants. Finally the usefulness of these higher order spectra is illustrated in the last chapter: for a non-Gaussian process (for which at least one higher order spectrum is non zero) the transfer function can be identified without providing any phase conditions. This is not the case for Gaussian processes where identifiability requires a parametric restriction of the ”minimum phase type”. This property allows to capture the generating noise of the process by deconvolution for example with seismic data.

This text is at the level of a postgraduate university course. It constitutes a useful reference on complementary probabilistic or statistical results for any person specialized in time series who wishes to learn more about the mathematical background. The techniques used for deriving limit theorems are ”non ordered” and the concept of martingale is not used. This allows the author to lead us gently into stationary random fields for which classical results in dimension one are extended in a natural way: central limit theorem, spectral estimation, Whittle’s parametric model estimation. Both time and frequency domain approaches are developed.

Let us outline two new topics discussed in this book. The C.L.T. for a field (or a sequence) is obtained with the following hypothesis: a strongly mixing field and the convergence to infinity of the \(L^{2+\delta}\) norm of the sums. Hence no conditions are imposed on the rate of decrease of the mixing coefficients and the techniques used, which are free of any ordering, are independent of the dimension k of the domain \(Z^ k\) on which the process is defined.

This C.L. result allows to prove subsequently the convergence to a normal distribution of the empirical covariances, of the smoothed spectrogram and of Whittle’s estimator for a field in \(L^ 8\) with summable cumulants of order 4. Let us note nevertheless an important difference between the cases \(k=1\) and \(k\geq 2\), a difference which is not pointed out by the author: because of edge effects the proposed estimators are \(n^{k/2}\) biased if \(k>1\) and the domain of observation is the hypercube of side n. To counter this, data on edges need to be tapered.

The second new topic which is approached concerns the study of non- Gaussian linear processes. First, a whole chapter is devoted to the estimation of the spectral density of the cumulants. Finally the usefulness of these higher order spectra is illustrated in the last chapter: for a non-Gaussian process (for which at least one higher order spectrum is non zero) the transfer function can be identified without providing any phase conditions. This is not the case for Gaussian processes where identifiability requires a parametric restriction of the ”minimum phase type”. This property allows to capture the generating noise of the process by deconvolution for example with seismic data.

Reviewer: X.Guyon

### MSC:

62M15 | Inference from stochastic processes and spectral analysis |

60G10 | Stationary stochastic processes |

60G60 | Random fields |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

60F05 | Central limit and other weak theorems |

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |