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We consider the notion of the matrix (tensor) distribution of a measurable function of several variables. On the one hand, this is an invariant of this function with respect to a certain group of transformations of variables; on the other hand, this is a special probability measure in the space of matrices (tensors) that is invariant under actions of natural infinite permutation groups. The intricate interplay of both interpretations of matrix (tensor) distributions makes them an important subject of modern functional analysis. We formulate and prove a theorem that, under certain conditions on a measurable function of two variables, its matrix distribution is a complete invariant. [ABSTRACT FROM AUTHOR]

An increasing sequence (an) of positive integers which satisfies an+1/an ≥ 1+ŋ for some positive ŋ is called a lacunary sequence. It has been known for over twenty years that every lacunary sequence has the strong sweeping out property which means that in every aperiodic dynamical sys¬tem we can find a set E of arbitrary small measure so that ... and ... almost everywhere. In this paper, we improve this result by showing that if (an) satisfies only ... for some positive ŋ,, then it already has the strong sweeping out property. [ABSTRACT FROM AUTHOR]

We present new versions of the CLT which are valid for each ergodic homogeneous multivariate random field (X 1 (⋅) , ... , X d (⋅)) on R m or Z m (m ≥ 1) (in particular, for each ergodic stationary random process and for each ergodic stationary random sequence) such that for all l E [ | X l (0) | 2 + δ ] < ∞ for some δ > 0 (l = 1 ,... , d) ; in some statements ergodicity is not assumed. Randomization made it possible to significantly weaken the strong mixing conditions and other restrictions of dependence, that are imposed in the conventional CLTs. These results pave the way to consistent statistical inference for homogeneous random fields and stationary processes with strong dependence. [ABSTRACT FROM AUTHOR]