scholarly journals Two special subgroups of the universal sofic group

2018 ◽  
Vol 39 (12) ◽  
pp. 3250-3261
Author(s):  
MATTEO CAVALERI ◽  
RADU B. MUNTEANU ◽  
LIVIU PĂUNESCU
Keyword(s):  
Probability Space ◽  
Group Of Automorphisms ◽  
Abelian Subalgebra ◽  
Standard Probability

We define a subgroup of the universal sofic group, obtained as the normalizer of a separable abelian subalgebra. This subgroup can be obtained as an extension by the group of automorphisms on a standard probability space. We show that each sofic representation can be conjugated inside this subgroup.

scholarly journals Entropy of systems

2016 ◽  
Vol 38 (3) ◽  
pp. 1118-1126
Author(s):  
RADU B. MUNTEANU
Keyword(s):  
Positive Integer ◽  
Probability Space ◽  
Bernoulli Shift ◽  
Ergodic Measure ◽  
Measure Preserving Transformation ◽  
Zero Entropy ◽  
Standard Probability

In this paper we show that any ergodic measure preserving transformation of a standard probability space which is $\text{AT}(n)$ for some positive integer $n$ has zero entropy. We show that for every positive integer $n$ any Bernoulli shift is not $\text{AT}(n)$. We also give an example of a transformation which has zero entropy but does not have property $\text{AT}(n)$ for any integer $n\geq 1$.

Positive Lyapunov exponents for a dense set of bounded measurable SL(2, ℝ)-cocycles

1992 ◽  
Vol 12 (2) ◽  
pp. 319-331 ◽  
Author(s):  
Oliver Knill
Keyword(s):  
Lyapunov Exponents ◽  
Probability Space ◽  
Image Position ◽  
Dense Set ◽  
Standard Probability

AbstractLet T be an aperiodic automorphism of a standard probability space (X, m). We prove, that the setis dense in L∞(X, SL(2, ℝ)).

scholarly journals Groups with infinite FC-center have the Schmidt property

2021 ◽  
pp. 1-46
Author(s):  
YOSHIKATA KIDA ◽  
ROBIN TUCKER-DROB
Keyword(s):  
Probability Space ◽  
Amenable Group ◽  
Ergodic Measure ◽  
Countable Group ◽  
Full Group ◽  
Orbit Equivalence ◽  
Central Sequence ◽  
Property T ◽  
Finite Conjugacy ◽  
Standard Probability

Abstract We show that every countable group with infinite finite conjugacy (FC)-center has the Schmidt property, that is, admits a free, ergodic, measure-preserving action on a standard probability space such that the full group of the associated orbit equivalence relation contains a non-trivial central sequence. As a consequence, every countable, inner amenable group with property (T) has the Schmidt property.

scholarly journals Some results on the span of families of Banach valued independent, random variables

1991 ◽  
Vol 14 (2) ◽  
pp. 381-384
Author(s):  
Rohan Hemasinha
Keyword(s):  
Banach Space ◽  
Probability Space ◽  
Linear Span ◽  
Random Variables ◽  
Isomorphic Copy ◽  
Independent Random Variables ◽  
Closed Linear Span

LetEbe a Banach space, and let(Ω,ℱ,P)be a probability space. IfL1(Ω)contains an isomorphic copy ofL1[0,1]then inLEP(Ω)(1≤P<∞), the closed linear span of every sequence of independent,Evalued mean zero random variables has infinite codimension. IfEis reflexive orB-convex and1<P<∞then the closed(in LEP(Ω))linear span of any family of independent,Evalued, mean zero random variables is super-reflexive.

scholarly journals On Some Conditions for Strong Law of Large Numbers for Weighted Sums of END Random Variables under Sublinear Expectations

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Xiaochen Ma ◽  
Qunying Wu
Keyword(s):  
Probability Space ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Weighted Sums ◽  
Dependent Random Variables ◽  
Strong Law ◽  
Sublinear Expectation ◽  
Negatively Dependent Random Variables ◽  
Large Numbers ◽  
End Random Variables

In this article, we research some conditions for strong law of large numbers (SLLNs) for weighted sums of extended negatively dependent (END) random variables under sublinear expectation space. Our consequences contain the Kolmogorov strong law of large numbers and the Marcinkiewicz strong law of large numbers for weighted sums of extended negatively dependent random variables. Furthermore, our results extend strong law of large numbers for some sequences of random variables from the traditional probability space to the sublinear expectation space context.

scholarly journals L2-BETTI NUMBERS OF DISCRETE MEASURED GROUPOIDS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 1169-1188 ◽  
Author(s):  
ROMAN SAUER
Keyword(s):  
Probability Space ◽  
Betti Numbers ◽  
Free Action ◽  
Countable Group ◽  
Equivalence Relations ◽  
Dimension Theory ◽  
Discrete Groups ◽  
Type Ii ◽  
Orbit Equivalent ◽  
Definition Of

There are notions of L2-Betti numbers for discrete groups (Cheeger–Gromov, Lück), for type II1-factors (recent work of Connes-Shlyakhtenko) and for countable standard equivalence relations (Gaboriau). Whereas the first two are algebraically defined using Lück's dimension theory, Gaboriau's definition of the latter is inspired by the work of Cheeger and Gromov. In this work we give a definition of L2-Betti numbers of discrete measured groupoids that is based on Lück's dimension theory, thereby encompassing the cases of groups, equivalence relations and holonomy groupoids with an invariant measure for a complete transversal. We show that with our definition, like with Gaboriau's, the L2-Betti numbers [Formula: see text] of a countable group G coincide with the L2-Betti numbers [Formula: see text] of the orbit equivalence relation [Formula: see text] of a free action of G on a probability space. This yields a new proof of the fact the L2-Betti numbers of groups with orbit equivalent actions coincide.

Stochastic comparisons of order statistics under multivariate imperfect repair

1996 ◽  
Vol 33 (1) ◽  
pp. 156-163 ◽  
Author(s):  
Taizhong Hu
Keyword(s):  
Order Statistics ◽  
Probability Space ◽  
Random Variables ◽  
Stochastic Comparisons ◽  
Special Model ◽  
Imperfect Repair ◽  
Monotone Coupling ◽  
Common Probability

A monotone coupling of order statistics from two sets of independent non-negative random variables Xi, i = 1, ···, n, and Yi, i = 1, ···, n, means that there exist random variables X′i, Y′i, i = 1, ···, n, on a common probability space such that , and a.s. j = 1, ···, n, where X(1) ≦ X(2) ≦ ·· ·≦ X(n) are the order statistics of Xi, i = 1, ···, n (with the corresponding notations for the X′, Y, Y′ sample). In this paper, we study the monotone coupling of order statistics of lifetimes in two multi-unit systems under multivariate imperfect repair. Similar results for a special model due to Ross are also given.

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