The existence of measures of a given cocycle, II: probability measures

2008 ◽  
Vol 28 (5) ◽  
pp. 1615-1633 ◽  
Author(s):  
BENJAMIN MILLER
Keyword(s):  
Probability Measure ◽  
Equivalence Relation ◽  
Polish Space ◽  
Probability Measures ◽  
Borel Equivalence Relation ◽  
Almost Everywhere

AbstractGiven a Polish space X, a countable Borel equivalence relation E on X, and a Borel cocycle $\rho : E \rightarrow (0, \infty )$, we characterize the circumstances under which there is a probability measure μ on X such that ρ(ϕ−1(x),x)=[d(ϕ*μ)/dμ](x) μ-almost everywhere, for every Borel injection ϕ whose graph is contained in E.

scholarly journals Incomparable actions of free groups

2016 ◽  
Vol 37 (7) ◽  
pp. 2084-2098
Author(s):  
CLINTON T. CONLEY ◽  
BENJAMIN D. MILLER
Keyword(s):  
Probability Measure ◽  
Free Group ◽  
Equivalence Relation ◽  
Polish Space ◽  
Borel Probability Measure ◽  
Free Groups ◽  
Borel Equivalence Relation ◽  
Separability Condition ◽  
Borel Probability ◽  
Free Actions

Suppose that $X$ is a Polish space, $E$ is a countable Borel equivalence relation on $X$, and $\unicode[STIX]{x1D707}$ is an $E$-invariant Borel probability measure on $X$. We consider the circumstances under which for every countable non-abelian free group $\unicode[STIX]{x1D6E4}$, there is a Borel sequence $(\cdot _{r})_{r\in \mathbb{R}}$ of free actions of $\unicode[STIX]{x1D6E4}$ on $X$, generating subequivalence relations $E_{r}$ of $E$ with respect to which $\unicode[STIX]{x1D707}$ is ergodic, with the further property that $(E_{r})_{r\in \mathbb{R}}$ is an increasing sequence of relations which are pairwise incomparable under $\unicode[STIX]{x1D707}$-reducibility. In particular, we show that if $E$ satisfies a natural separability condition, then this is the case as long as there exists a free Borel action of a countable non-abelian free group on $X$, generating a subequivalence relation of $E$ with respect to which $\unicode[STIX]{x1D707}$ is ergodic.

The existence of measures of a given cocycle, I: atomless, ergodic σ-finite measures

2008 ◽  
Vol 28 (5) ◽  
pp. 1599-1613 ◽  
Author(s):  
BENJAMIN MILLER
Keyword(s):  
Equivalence Relation ◽  
Polish Space ◽  
Finite Measure ◽  
Borel Equivalence Relation ◽  
Almost Everywhere ◽  
Finite Measures

AbstractGiven a Polish space X, a countable Borel equivalence relation E on X, and a Borel cocycle $\rho : E \rightarrow (0, \infty )$, we characterize the circumstances under which there is a suitably non-trivial σ-finite measure μ on X such that, for every Borel injection ϕ whose graph is contained in E, ρ(ϕ−1(x),x)=[d(ϕ*μ)/dμ](x) μ-almost everywhere.

scholarly journals Amenable versus hyperfinite Borel equivalence relations

1993 ◽  
Vol 58 (3) ◽  
pp. 894-907 ◽  
Author(s):  
Alexander S. Kechris
Keyword(s):  
Probability Measure ◽  
Equivalence Relation ◽  
Borel Probability Measure ◽  
Amenable Group ◽  
Countable Group ◽  
Equivalence Relations ◽  
Borel Space ◽  
Borel Equivalence Relations ◽  
Borel Equivalence Relation ◽  
Standard Borel Space

LetXbe a standard Borel space (i.e., a Polish space with the associated Borel structure), and letEbe acountableBorel equivalence relation onX, i.e., a Borel equivalence relationEfor which every equivalence class [x]Eis countable. By a result of Feldman-Moore [FM],Eis induced by the orbits of a Borel action of a countable groupGonX.The structure of general countable Borel equivalence relations is very little understood. However, a lot is known for the particularly important subclass consisting of hyperfinite relations. A countable Borel equivalence relation is calledhyperfiniteif it is induced by a Borel ℤ-action, i.e., by the orbits of a single Borel automorphism. Such relations are studied and classified in [DJK] (see also the references contained therein). It is shown in Ornstein-Weiss [OW] and Connes-Feldman-Weiss [CFW] that for every Borel equivalence relationEinduced by a Borel action of a countable amenable groupGonXand for every (Borel) probability measure μ onX, there is a Borel invariant setY⊆Xwith μ(Y) = 1 such thatE↾Y(= the restriction ofEtoY) is hyperfinite. (Recall that a countable group G isamenableif it carries a finitely additive translation invariant probability measure defined on all its subsets.) Motivated by this result, Weiss [W2] raised the question of whether everyEinduced by a Borel action of a countable amenable group is hyperfinite. Later on Weiss (personal communication) showed that this is true forG= ℤn. However, the problem is still open even for abelianG. Our main purpose here is to provide a weaker affirmative answer for general amenableG(and more—see below). We need a definition first. Given two standard Borel spacesX, Y, auniversally measurableisomorphism betweenXandYis a bijection ƒ:X→Ysuch that both ƒ, ƒ-1are universally measurable. (As usual, a mapg:Z→W, withZandWstandard Borel spaces, is calleduniversally measurableif it is μ-measurable for every probability measure μ onZ.) Notice now that to assert that a countable Borel equivalence relation onXis hyperfinite is trivially equivalent to saying that there is a standard Borel spaceYand a hyperfinite Borel equivalence relationFonY, which isBorelisomorphic toE, i.e., there is a Borel bijection ƒ:X→YwithxEy⇔ ƒ(x)Fƒ(y). We have the following theorem.

Equivalence singularity dichotomies for a class of ergodic measures

1977 ◽  
Vol 81 (2) ◽  
pp. 249-252 ◽  
Author(s):  
Marek Kanter
Keyword(s):  
Probability Measure ◽  
Equivalence Relation ◽  
Borel Subset ◽  
Ergodic Measure ◽  
Unit Mass ◽  
Probability Measures ◽  
Borel Sets ◽  
Ergodic Measures ◽  
Countable Subgroup ◽  
Two Measures

Let µ be a probability measure on the Borel subsets of R∞. If D is a countable subgroup of R∞ we say that µ is D-ergodic if (1) for any D invariant Borel subset A of R we have µ(A) = 0 or 1 and (2) if µ*δx ≈ µ for all x ∈ D (where δx stands for unit mass at x while the equivalence relation ≈ signifies that the two measures have the same null sets.) We say that x is an admissible translate for µ if µ*δx ≈ µ. We say that µ is D-smooth if sx is an admissible translate for µ for all x ∈ D and all s ∈ R. We say that µ is a smooth ergodic measure if µ is D-ergodic and D-smooth for some countable subgroup D as above. In this paper we show that any two smooth ergodic probability measures µl, µ2 are either equivalent or singular (where the latter means that there exist disjoint Borel sets Al, A2 ⊂ R∞ such that µi(Ai) = 1 and is signified by µ1 ┴ µ2). It is important to note that the countable subgroup D1 associated with µl need not be the same as the subgroup D2 associated with µ2.

scholarly journals A note on the idempotent measures on countable semigroups

1970 ◽  
Vol 11 (4) ◽  
pp. 417-420
Author(s):  
Tze-Chien Sun ◽  
N. A. Tserpes
Keyword(s):  
Probability Measure ◽  
Continuous Mapping ◽  
Compact Semigroup ◽  
Probability Measures ◽  
Left Zero ◽  
Product Topology ◽  
Locally Compact ◽  
Locally Compact Semigroup ◽  
Image Position ◽  
Completely Simple

In [6] we announced the following Conjecture: Let S be a locally compact semigroup and let μ be an idempotent regular probability measure on S with support F. Then(a) F is a closed completely simple subsemigroup.(b) F is isomorphic both algebraically and topologically to a paragroup ([2], p.46) X × G × Y where X and Y are locally compact left-zero and right-zero semi-groups respectively and G is a compact group. In X × G × Y the topology is the product topology and the multiplication of any two elements is defined by , x where [y, x′] is continuous mapping from Y × X → G.(c) The induced μ on X × G × Y can be decomposed as a product measure μX × μG× μY where μX and μY are two regular probability measures on X and Y respectively and μG is the normed Haar measure on G.

scholarly journals Countable sections for locally compact group actions

1992 ◽  
Vol 12 (2) ◽  
pp. 283-295 ◽  
Author(s):  
Alexander S. Kechris
Keyword(s):  
Compact Group ◽  
Equivalence Relation ◽  
Group Actions ◽  
Locally Compact Group ◽  
Borel Space ◽  
Locally Compact ◽  
Borel Equivalence Relation ◽  
Measurable Section ◽  
Singular Action

AbstractIt has been shown by J. Feldman, P. Hahn and C. C. Moore that every non-singular action of a second countable locally compact group has a countable (in fact so-called lacunary) complete measurable section. This is extended here to the purely Borel theoretic category, consisting of a Borel action of such a group on an analytic Borel space (without any measure). Characterizations of when an arbitrary Borel equivalence relation admits a countable complete Borel section are also established.

Central Limit Theorems for Interchangeable Processes

1958 ◽  
Vol 10 ◽  
pp. 222-229 ◽  
Author(s):  
J. R. Blum ◽  
H. Chernoff ◽  
M. Rosenblatt ◽  
H. Teicher
Keyword(s):  
Stochastic Process ◽  
Probability Measure ◽  
Central Limit ◽  
Joint Distribution ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Probability Measures ◽  
Image Position ◽  
Positive Integers

Let {Xn} (n = 1, 2 , …) be a stochastic process. The random variables comprising it or the process itself will be said to be interchangeable if, for any choice of distinct positive integers i 1, i 2, H 3 … , ik, the joint distribution of depends merely on k and is independent of the integers i 1, i 2, … , i k. It was shown by De Finetti (3) that the probability measure for any interchangeable process is a mixture of probability measures of processes each consisting of independent and identically distributed random variables.

scholarly journals Robust decision making and convex risk measures computation

2015 ◽  
Author(s):  
Γεώργιος Παπαγιάννης
Keyword(s):  
Decision Making ◽  
Probability Measure ◽  
Calculus Of Variations ◽  
Model Uncertainty ◽  
Risk Measures ◽  
Risk Measure ◽  
Probability Measures ◽  
Discrete Measure ◽  
Convex Risk Measures ◽  
Variance Risk

The main aim of the present thesis is to investigate the effect of diverging priors concerning model uncertainty on decision making. One of the main issues in the thesis is to assess the effect of different notions of distance in the space of probability measures and their use as loss functionals in the process of identifying the best suited model among a set of plausible priors. Another issue, is that of addressing the problem of ``inhomogeneous" sets of priors, i.e. sets of priors that highly divergent opinions may occur, and the need to robustly treat that case. As high degrees of inhomogeneity may lead to distrust of the decision maker to the priors it may be desirable to adopt a particular prior corresponding to the set which somehow minimizes the ``variability" among the models on the set. This leads to the notion of Frechet risk measure. Finally, an important problem is the actual calculation of robust risk measures. An account of their variational definition, the problem of calculation leads to the numerical treatment of problems of the calculus of variations for which reliable and effective algorithms are proposed. The contributions of the thesis are presented in the following three chapters. In Chapter 2, a statistical learning scheme is introduced for constructing the best model compatible with a set of priors provided by different information sources of varying reliability. As various priors may model well different aspects of the phenomenon the proposed scheme is a variational scheme based on the minimization of a weighted loss function in the space of probability measures which in certain cases is shown to be equivalent to weighted quantile averaging schemes. Therefore in contrast to approaches such as minimax decision theory in which a particular element of the prior set is chosen we construct for each prior set a probability measure which is not necessarily an element of it, a fact that as shown may lead to better description of the phenomenon in question. While treating this problem we also address the issue of the effect of the choice of distance functional in the space of measures on the problem of model selection. One of the key findings in this respect is that the class of Wasserstein distances seems to have the best performance as compared to other distances such as the KL-divergence. In Chapter 3, motivated by the results of Chapter 2, we treat the problem of specifying the risk measure for a particular loss when a set of highly divergent priors concerning the distribution of the loss is available. Starting from the principle that the ``variability" of opinions is not welcome, a fact for which a strong axiomatic framework is provided (see e.g. Klibanoff (2005) and references therein) we introduce the concept of Frechet risk measures, which corresponds to a minimal variance risk measure. Here we view a set of priors as a discrete measure on the space of probability measures and by variance we mean the variance of this discrete probability measure. This requires the use of the concept of Frechet mean. By different metrizations of the space of probability measures we define a variety of Frechet risk measures, the Wasserstein, the Hellinger and the weighted entropic risk measure, and illustrate their use and performance via an example related to the static hedging of derivatives under model uncertainty. In Chapter 4, we consider the problem of numerical calculation of convex risk measures applying techniques from the calculus of variations. Regularization schemes are proposed and the theoretical convergence of the algorithms is considered.

The Gaussian distribution revisited

1996 ◽  
Vol 28 (2) ◽  
pp. 500-524 ◽  
Author(s):  
Carlos E. Puente ◽  
Miguel M. López ◽  
Jorge E. Pinzón ◽  
José M. Angulo
Keyword(s):  
Fractal Dimension ◽  
Probability Measure ◽  
Gaussian Distribution ◽  
Closed Interval ◽  
Fractal Dimensions ◽  
Cumulative Distribution ◽  
Probability Measures ◽  
New Construction

A new construction of the Gaussian distribution is introduced and proven. The procedure consists of using fractal interpolating functions, with graphs having increasing fractal dimensions, to transform arbitrary continuous probability measures defined over a closed interval. Specifically, let X be any probability measure on the closed interval I with a continuous cumulative distribution. And let fΘ,D:I → R be a deterministic continuous fractal interpolating function, as introduced by Barnsley (1986), with parameters Θ and fractal dimension for its graph D. Then, the derived measure Y = fΘ,D(X) tends to a Gaussian for all parameters Θ such that D → 2, for all X. This result illustrates that plane-filling fractal interpolating functions are ‘intrinsically Gaussian'. It explains that close approximations to the Gaussian may be obtained transforming any continuous probability measure via a single nearly-plane filling fractal interpolator.

The Gaussian distribution revisited

1996 ◽  
Vol 28 (02) ◽  
pp. 500-524 ◽  
Author(s):  
Carlos E. Puente ◽  
Miguel M. López ◽  
Jorge E. Pinzón ◽  
José M. Angulo
Keyword(s):  
Fractal Dimension ◽  
Probability Measure ◽  
Gaussian Distribution ◽  
Closed Interval ◽  
Fractal Dimensions ◽  
Cumulative Distribution ◽  
Probability Measures ◽  
New Construction

A new construction of the Gaussian distribution is introduced and proven. The procedure consists of using fractal interpolating functions, with graphs having increasing fractal dimensions, to transform arbitrary continuous probability measures defined over a closed interval. Specifically, let X be any probability measure on the closed interval I with a continuous cumulative distribution. And let f Θ,D :I → R be a deterministic continuous fractal interpolating function, as introduced by Barnsley (1986), with parameters Θ and fractal dimension for its graph D. Then, the derived measure Y = f Θ,D (X) tends to a Gaussian for all parameters Θ such that D → 2, for all X. This result illustrates that plane-filling fractal interpolating functions are ‘intrinsically Gaussian'. It explains that close approximations to the Gaussian may be obtained transforming any continuous probability measure via a single nearly-plane filling fractal interpolator.

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