### A characterization and moving average representation for stable harmonizable processes.

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Let $\left\{{X}_{n}\right\}$ be a stationary and ergodic time series taking values from a finite or countably infinite set $\mathcal{X}$ and that $f\left(X\right)$ is a function of the process with finite second moment. Assume that the distribution of the process is otherwise unknown. We construct a sequence of stopping times ${\lambda}_{n}$ along which we will be able to estimate the conditional expectation $E\left(f\left({X}_{{\lambda}_{n}+1}\right)\right|{X}_{0},\cdots ,{X}_{{\lambda}_{n}})$ from the observations $({X}_{0},\cdots ,{X}_{{\lambda}_{n}})$ in a point wise consistent way for a restricted class of stationary and ergodic finite or countably infinite alphabet time series...

The uniqueness of the Wold decomposition of a finite-dimensional stationary process without assumption of full rank stationary process and the Lebesgue decomposition of its spectral measure is easily obtained.

Let ${X}_{mn}$ be a second-order stationary random field on Z². Let ℳ(L) be the linear span of ${X}_{mn}:m\le 0,n\in Z$, and ℳ(RN) the linear span of ${X}_{mn}:m\ge N,n\in Z$. Spectral criteria are given for the condition $li{m}_{N\to \infty}{c}_{N}=0$, where ${c}_{N}$ is the cosine of the angle between ℳ(L) and $\mathcal{M}\left({R}_{N}\right)$.

There are two kinds of universal schemes for estimating residual waiting times, those where the error tends to zero almost surely and those where the error tends to zero in some integral norm. Usually these schemes are different because different methods are used to prove their consistency. In this note we will give a single scheme where the average error is eventually small for all time instants, while the error itself tends to zero along a sequence of stopping times of density one.

Let $\mathbf{X}$ and $\mathbf{Y}$ be stationarily cross-correlated multivariate stationary sequences. Assume that all values of $\mathbf{Y}$ and all but one values of $\mathbf{X}$ are known. We determine the best linear interpolation of the unknown value on the basis of the known values and derive a formula for the interpolation error matrix. Our assertions generalize a result of Budinský [1].

Important characteristics of any algorithm are its complexity and speed in real calculations. From this point of view, we analyze some algorithms for prediction in finite stationary time series. First, we review results developed by P. Bondon [1] and then, we derive the complexities of Levinson and a new algorithm. It is shown that the time needed for real calculations of predictions is proportional to the theoretical complexity of the algorithm. Some practical recommendations for the selection...