Robert Azencott

Random Fields and Stochastic Integrals

We shall see in Chapter 6 that every stationary process is the Fourier transform of a random measure carried by ∏= [−π,+ π[. We describe here the uncorrelated random fields which formalize the intuitive notion of random measure. The theory extends easily to more general spaces ∏.

Nonstationary ARMA Processes and Forecasting

To modelize a nonstationary random phenomenon, it is tempting to write Yn = f(n) + Xn where X is stationary and the trend f(n) is a deterministic function to be adequately selected; f(n) is often assumed to be the sum of a polynomial in n and of linear combinations of cos nλj and sin nλj,= 1,2, ..., K. A first approach is to estimate separately f (for instance by least squares methods (cf. [15]) and the covariance structure of X.

Discrete Time Random Processes

The experimental description of any random phenomenon involves a family of numbers Xt, t ɛ T. Since Kolmogorov, it has been mathematically convenient to summarize the impact of randomness through the stochastic choice of a point in an adequate set Ω (space of trials) and to consider the random variables Xt as well determined functions on Ω with values in ℝ.

Spectral Representation of Stationary Processes

Since we may identify the one dimensional torus with TT = [-π,π,[to each bounded measure v TT on we associate its Fourier transform

Empirical Estimators and Periodograms

\hat \nu (n) = \int_{TT} {{e^{in\lambda }}d\nu \left( \lambda \right),\;n \in ZZ}

Identification and Compensated Likelihood

Let X be a stationary, centered, nonzero gaussian process, with spectral density f. Let μn be the law of X1…Xn, hn the density of μn on ℝn and L (f, X1…Xn) = log hn (X1…Xn) the log-likelihood of X. We have seen (Chapter 12, Section 1.2) that