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A probabilistic ergodic decomposition result

Albert Raugi (2009)

Annales de l'I.H.P. Probabilités et statistiques

Let ( X , 𝔛 , μ ) be a standard probability space. We say that a sub-σ-algebra 𝔅 of 𝔛 decomposes μ in an ergodic way if any regular conditional probability 𝔅 P with respect to 𝔅 andμ satisfies, for μ-almost every x∈X, B 𝔅 , 𝔅 P ( x , B ) { 0 , 1 } . In this case the equality μ ( · ) = X 𝔅 P ( x , · ) μ ( d x ) , gives us an integral decomposition in “ 𝔅 -ergodic” components. For any sub-σ-algebra 𝔅 of 𝔛 , we denote by 𝔅 ¯ the smallest sub-σ-algebra of 𝔛 containing 𝔅 and the collection of all setsAin 𝔛 satisfyingμ(A)=0. We say that 𝔅 isμ-complete if 𝔅 = 𝔅 ¯ . Let { 𝔅 i i I } be a non-empty family...

About the density of spectral measure of the two-dimensional SaS random vector

Marta Borowiecka-Olszewska, Jolanta K. Misiewicz (2003)

Discussiones Mathematicae Probability and Statistics

In this paper, we consider a symmetric α-stable p-sub-stable two-dimensional random vector. Our purpose is to show when the function e x p - ( | a | p + | b | p ) α / p is a characteristic function of such a vector for some p and α. The solution of this problem we can find in [3], in the language of isometric embeddings of Banach spaces. Our proof is based on simple properties of stable distributions and some characterization given in [4].

An Alpern tower independent of a given partition

James T. Campbell, Jared T. Collins, Steven Kalikow, Raena King, Randall McCutcheon (2015)

Colloquium Mathematicae

Given a measure-preserving transformation T of a probability space (X,ℬ,μ) and a finite measurable partition ℙ of X, we show how to construct an Alpern tower of any height whose base is independent of the partition ℙ. That is, given N ∈ ℕ, there exists a Rokhlin tower of height N, with base B and error set E, such that B is independent of ℙ, and TE ⊂ B.

Bernoulli sequences and Borel measurability in ( 0 , 1 )

Petr Veselý (1993)

Commentationes Mathematicae Universitatis Carolinae

The necessary and sufficient condition for a function f : ( 0 , 1 ) [ 0 , 1 ] to be Borel measurable (given by Theorem stated below) provides a technique to prove (in Corollary 2) the existence of a Borel measurable map H : { 0 , 1 } { 0 , 1 } such that ( H ( X p ) ) = ( X 1 / 2 ) holds for each p ( 0 , 1 ) , where X p = ( X 1 p , X 2 p , ... ) denotes Bernoulli sequence of random variables with P [ X i p = 1 ] = p .

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