Abolghassem Miamee

Periodically correlated processes and their stationary dilations

An explicit form of a stationary dilation for periodically correlated random sequences is obtained. The spectrum of this dilation sequence is related to that of the original periodically correlated sequence in a fairly simple manner. These results are then used to give spectral conditions for a periodically correlated sequence to be deterministic, purely nondeterministic, minimal, and to have a positive angle between its past and future.

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.

Continuous time periodically correlated processes: Spectrum and prediction

In this paper a characterization of the spectrum and the random spectrum of a bounded continuous parameter periodically correlated process is given. It is shown that with any bounded periodically correlated process one can associate an appropriate infinite dimensional stationary process which shares its regularity properties. This stationary process is used to obtain Wold decomposition and a regularity condition for a periodically correlated process.

Regularity and Minimality of Infinite Variance Processes

For stationary processes with infinite variance the notions of covariance and spectrum are not defined. We characterize regularity and minimality of such processes in the spirit of some classical results for second-order processes, namely values of the process forming conditional basis for their spans. Several open problems are discussed.

On the geometry of L p (μ) with applications to infinite variance processes

Some geometric properties of L p spaces are studied which shed light on the prediction of infinite variance processes. In particular, a Pythagorean theorem for L p is derived. Improved growth rates for the moving average parameters are obtained.

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 Spectral Domain of Periodically Correlated Processes

The content of this article primarily falls into three sections. Section 1 deals with a basic structural spectral representation theorem for periodically correlated sequences. Section 2 provides a certain class of square integrable functions that is isomorphic to the time domain of the sequence. This complete class is called the spectral domain of the sequence. Section 3 is related to the unsuccessful previous attempts by others to construct a complete function space as the spectral domain. These are natural extensions of the known results for the stationary case to the periodically correlated sequences.

On the spectrum of correlation autoregressive sequences

In this paper some properties of the correlation autoregressive (CAR) sequences are studied. A representation for the correlation function of an arbitrary CAR sequence is obtained and the relationship between a CAR equation and the growth of the variance and location of spectral lines is revealed. It is also observed that bounded correlation autoregressive sequences coincide with almost periodically correlated sequences with the spectral measure supported on finitely many lines. As a consequence a characterization of the spectrum of a bounded CAR sequence is provided.

Positive operators on a Banach space and the Fejér-Riesz theorem

The Fejer and Riesz theorem on the factorization of nonnega- tive trigonometric polynomials is extended to the nonnegative operator valued trigonometric polynomials on a Banach space. The work is based on the analysis of quasi square roots of nonnegative operator valued functions on a Banach space.