Let $\left\{W_t\right\}=\left\{{(X_{t^{\text{'}}}^{\text{'}},Y_t^{\text{'}})}^{\text{'}}\right\}$ be vector ARMA $(m,n)$ processes. Denote by ${\widehat{X}}_t\left(a\right)$ the predictor of $X_t$ based on $X_{t-a},X_{t-a-1},...$ and by ${\widehat{X}}_t(a,b)$ the predictor of $X_t$ based on $X_{t-a},X_{t-a-1},...,Y_{t-b},Y_{t-b-1},...$. The accuracy of the predictors is measured by ${\Delta }_X\left(a\right)=\text{E}[X_t-{\widehat{X}}_t\left(a\right)]{[X_t-{\widehat{X}}_t\left(a\right)]}^{\text{'}}$ and ${\Delta }_X(a,b)=\text{E}[X_t-{\widehat{X}}_t(a,b)]{[X_t-{\widehat{X}}_t(a,b)]}^{\text{'}}$. A general sufficient condition for the equality ${\Delta }_X\left(a\right)={\Delta }_X(a,a)]$ is given in the paper and it is shown that the equality ${\Delta }_X\left(1\right)={\Delta }_X(1,1)]$ implies ${\Delta }_X\left(a\right)={\Delta }_X(a,a)]$ for all natural numbers $a$.

A one-to-one correspondence between locally square integrable periodically correlated (PC) processes and a certain class of infinite-dimensional stationary processes is obtained. The correspondence complements and clarifies Gladyshev's known result [3] describing the correlation function of a continuous periodically correlated process. In contrast to Gladyshev's paper, the procedure for explicit reconstruction of one process from the other is provided. A representation of a PC process as a unitary...

For a binary stationary time series define ${\sigma }_n$ to be the number of consecutive ones up to the first zero encountered after time $n$, and consider the problem of estimating the conditional distribution and conditional expectation of ${\sigma }_n$ after one has observed the first $n$ outputs. We present a sequence of stopping times and universal estimators for these quantities which are pointwise consistent for all ergodic binary stationary processes. In case the process is a renewal process with zero the renewal state...

We give some estimation schemes for the conditional distribution and conditional expectation of the the next output following the observation of the first $n$ outputs of a stationary process where the random variables may take finitely many possible values. Our schemes are universal in the class of finitarily Markovian processes that have an exponential rate for the tail of the look back time distribution. In addition explicit rates are given. A necessary restriction is that the scheme proposes an...

Ce travail est une étude théorique d’opérateurs de Toeplitz dont le symbole est une fonction matricielle régulière définie positive partout sur le tore à une dimension. Nous proposons d’abord une formule d’inversion exacte pour un opérateur de Toeplitz à symbole matriciel, démontrée au moyen d’un théorème établi en annexe et donnant la solution du problème de la prédiction relatif à un passé fini pour un processus stationnaire du second ordre. Nous établissons ensuite, à partir de cet inverse, un...

In the first section of this paper there are given criteria for strict convexity and smoothness of the Bochner-Orlicz space with the Orlicz norm as well as the Luxemburg norm. In the second one that geometrical properties are applied to the characterization of metric projections and zero mean valued best approximants to Bochner-Orlicz spaces.

By a harmonizable sequence of random variables we mean the sequence of Fourier coefficients of a random measure M: $Xₙ\left(M\right)={\int }_0^1{e}^{2\pi nis}M\left(ds\right)$ (n = 0,±1,...) The paper deals with prediction problems for sequences Xₙ(M) for isotropic and atomless random measures M. The crucial result asserts that the space of all complex-valued M-integrable functions on the unit interval is a Musielak-Orlicz space. Hence it follows that the problem for Xₙ(M) (n = 0,±1,...) to be deterministic is in fact an extremal problem of Szegö’s type...

The knowledge of causal relations provides a possibility to perform predictions and helps to decide about the most reasonable actions aiming at the desired objectives. Although the causal reasoning appears to be natural for the human thinking, most of the traditional statistical methods fail to address this issue. One of the well-known methodologies correctly representing the relations of cause and effect is Pearl's causality approach. The paper brings an alternative, purely algebraic methodology...

We present a general method for the extension of results about linear prediction for q-variate weakly stationary processes on a separable locally compact abelian group $G_2$ (whose dual is a Polish space) with known values of the processes on a separable subset $S_2\subseteq G_2$ to results for weakly stationary processes on $G_1\times G_2$ with observed values on $G_1\times S_2$. In particular, the method is applied to obtain new proofs of some well-known results of Ze Pei Jiang.