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.

The Kaczmarz algorithm of successive projections suggests the following concept. A sequence $\left(e_k\right)$ of unit vectors in a Hilbert space is said to be effective if for each vector x in the space the sequence (xₙ) converges to x where (xₙ) is defined inductively: x₀ = 0 and $xₙ=x_{n-1}+\alpha ₙeₙ$, where $\alpha ₙ=⟨x-x_{n-1},eₙ⟩$. We prove the effectivity of some sequences in Hilbert spaces. We generalize the concept of effectivity to sequences of vectors in Banach spaces and we prove some results for this more general concept.

Rosenblatt showed that a stationary Gaussian random field is strongly mixing if it has a positive, continuous spectral density. In this article, spectral criteria are given for the rate of strong mixing in such a field.

An exact criterion is derived for an operator valued weight function $W(e^{is},e^{it})$ on the torus to have a factorization $W(e^{is},e^{it})=\Phi (e^{is},e^{it})*\Phi (e^{is},e^{it})$, where the operator valued Fourier coefficients of Φ vanish outside of the Helson-Lowdenslager halfplane $\Lambda =(m,n)\in {\mathbb{Z}}^2:m\ge 1\cup (0,n):n\ge 0$, and Φ is “outer” in a related sense. The criterion is expressed in terms of a regularity condition on the weighted space $L^2\left(W\right)$ of vector valued functions on the torus. A logarithmic integrability test is also provided. The factor Φ is explicitly constructed in terms of Toeplitz operators...