Andrzej Makagon

On AR(1) models with periodic and almost periodic coefficients

Abstract This paper is concerned with the existence of bounded solutions to the system of equations X n = a n X n −1 + ξ n , n ∈ Z , where ξ n are uncorrelated constant variance zero mean random variables. We give necessary and sufficient conditions for boundedness in the general case and then specifically for periodic and almost periodic ( a n ). This provides the first step in extending the periodic autoregressive models, for which boundedness is equivalent to the stationarity of the blocked vector sequence X k =(X kT ,X kT+1 ,…,X (k+1)T−1 ), to the almost periodic case.

On the hellinger square integral with respect to an operator valued measure and stationary processes

A construction of the Hellinger square integral with respect to a semispectral measure in a Banach space B is given. It is proved that the space of values of a B-valued stationary stochastic process is unitarily isomorphic to the space of all B*-valued measures that are Hellinger square integrable with respect to the spectral measure of the process. Some applications of the above theorem in the prediction theory (especially to interpolation problem) are also considered.

Structure of PC Sequences and the 3rd Prediction Problem

Founders of prediction theory formulated three prediction problems: extrapolation problem, interpolation problem, and the problem of positivity of the angle between the past and the future. The third one is strictly related to the question of representing the predictor as a series of past values of the process. All three have been solved in the case of stationary sequences, however, as far as we know in the case of periodically correlated sequences only the first prediction problem has been studied. The purpose of this chapter is to overview the third prediction problems.

On the support of the spectral measure of a harmonizable sequence

In this note we discuss a relationship between the correlation function of a harmonizable sequence and support of its spectral measure.

Periodically Correlated Sequences with Rational Spectra and PARMA Systems

An acronym PARMA is used in different configurations, we talk about PARMA systems, PARMA sequences, or PARMA models. This paper is a result of the author search to understand this complex world of PARMAs.