Murray Rosenblatt

A Note on Prediction and an Autoregressive Sequence

Prediction for a first order possibly nonGaus-sian sequence is considered. Remarks are made about prediction with time increasing and with time reversed.

Reversibility and Identifiability

Let us first consider linear stationary sequences. A sequence of independent, identically distributed real random variables ξj, j = …, -1,0,1,… is given with Eξj = 0, 0 < Eξ j 2 = σ2 < ∞. The process xj is obtained by passing this sequence through a linear filter characterized by the real weights, a j , ∑a j 2 < ∞, n n

Nongaussian autoregressive sequences and random fields

{x_{t}} = sumlimits_{{j = - infty }}^{infty } {{a_{j}}xi t - j.}

Spectral analysis and harmonizable processes

n n(1.1.1)

Minimum Phase Estimation

We consider the problem of estimating the filter generating a non-Gaussian linear process and the deconvolution of that process when the spectral density of the process has zeros. Without using a minimum phase assumption we show that often if there are only finitely many zeros there are procedures to effect such an estimation and deconvolution.

Homogeneous Gaussian Random Fields

In this paper we discuss estimation procedures for the parameters of autoregressive schemes. There is a large literature concerned with estimation in the one-dimensional Gaussian case. Much of our discussion will however be dedicated to the nonGaussian context, some aspects of which have been considered only in recent years. Results have also at times been obtained in the broader context of autoregressive moving average schemes. We restrict ourselves to the case of autoregressive schemes for the sake of simplicity. Also they are the discrete analogue of simple versions of stochastic differential equations with constant coefficients. It is also apparent that nonGaussian autoregressive stationary sequences have a richer and more complicated structure than that of the Gaussian autoregressive stationary sequences.

Cumulants, Mixing and Estimation for Gaussian Fields

Non-Gaussian linear time series models are discussed. The ways in which they differ from Gaussian models are noted. This is particularly the case for prediction and parameter or transfer function estimation.

The Fluctuation of the Quasi-Gaussian Likelihood

Spectral estimation for nonstationary harmonizable processes making use of a single realisation of the process as a function of time is considered. Consistent estimators based on smoothed versions of periodogram like forms are effective when the spectral support of the process consists of lines and the spectra are sufficiently smooth on the lines.