Standard Probability Space Research Articles

Complete convergence for weighted sums of widely negative dependent random variables under the sub-linear expectations

Using different research methods than probability space, this article investigates complete convergence for weighted sums of widely negative dependent random variables under sub-linear expectations. The major findings of this study are expansions of complete convergence for weighted sums of widely orthant-dependent random variables in the standard probability space.

The 𝐿²-completion of the space of Riemannian metrics is CAT(0): A shorter proof

We reprove in an easier way a result of Brian Clarke [Math. Z. 273 (2013), pp. 55–93]: the completion of the space of Riemannian metrics of a compact, orientable smooth manifold with respect to the L 2 L^2 -distance is CAT ( 0 ) (0) . In particular we show that this completion is isometric to the space of L 2 L^2 -maps from a standard probability space to a fixed CAT ( 0 ) (0) space.

Orbit equivalence rigidity of irreducible actions of right-angled Artin groups

Let $G_\Gamma \curvearrowright X$ and $G_\Lambda \curvearrowright Y$ be two free measure-preserving actions of one-ended right-angled Artin groups with trivial center on standard probability spaces. Assume they are irreducible, i.e. every element from a standard generating set acts ergodically. We prove that if the two actions are stably orbit equivalent (or merely stably $W^*$-equivalent), then they are automatically conjugate through a group isomorphism between $G_\Gamma$ and $G_\Lambda$. Through work of Monod and Shalom, we derive a superrigidity statement: if the action $G_\Gamma \curvearrowright X$ is stably orbit equivalent (or merely stably $W^*$-equivalent) to a free, measure-preserving, mildly mixing action of a countable group, then the two actions are virtually conjugate. We also use the works of Popa and Ioana, Popa and Vaes to establish the $W^*$-superrigidity of Bernoulli actions of all infinite conjugacy classes groups having a finite generating set made of infinite-order elements where two consecutive elements commute, and one has a nonamenable centralizer: these include one-ended nonabelian right-angled Artin groups, but also many other Artin groups and most mapping class groups of finite-type surfaces.

Existence and non-existence of solutions to the coboundary equation for measure-preserving systems

AbstractA fundamental question in the field of cohomology of dynamical systems is to determine when there are solutions to the coboundary equation: $$ \begin{align*} f = g - g \circ T. \end{align*} $$ In many cases, T is given to be an ergodic invertible measure-preserving transformation on a standard probability space $(X, {\mathcal B}, \mu )$ and is contained in $L^p$ for $p \geq 0$ . We extend previous results by showing for any measurable f that is non-zero on a set of positive measure, the class of measure-preserving T with a measurable solution g is meager (including the case where $\int _X f\,d\mu = 0$ ). From this fact, a natural question arises: given f, does there always exist a solution pair T and g? In regards to this question, our main results are as follows. Given measurable f, there exist an ergodic invertible measure-preserving transformation T and measurable function g such that $f(x) = g(x) - g(Tx)$ for almost every (a.e.) $x\in X$ , if and only if $\int _{f> 0} f\,d\mu = - \int _{f < 0} f\,d\mu $ (whether finite or $\infty $ ). Given mean-zero $f \in L^p(\mu )$ for $p \geq 1$ , there exist an ergodic invertible measure-preserving T and $g \in L^{p-1}(\mu )$ such that $f(x) = g(x) - g( Tx )$ for a.e. $x \in X$ . In some sense, the previous existence result is the best possible. For $p \geq 1$ , there exists a dense $G_{\delta }$ set of mean-zero $f \in L^p(\mu )$ such that for any ergodic invertible measure-preserving T and any measurable g such that $f(x) = g(x) - g(Tx)$ almost everywhere, then $g \notin L^q(\mu )$ for $q> p - 1$ . Finally, it is shown that we cannot expect finite moments for solutions g, when $f \in L^1(\mu )$ . In particular, given any such that $\lim _{x\to \infty } \phi (x) = \infty $ , there exist mean-zero $f \in L^1(\mu )$ such that for any solutions T and g, the transfer function g satisfies: $$ \begin{align*} \int_{X} \phi ( | g(x) | )\,d\mu = \infty. \end{align*} $$

Cocycle superrigidity from higher rank lattices to $ {{\rm{Out}}}{(F_N)} $

<p style='text-indent:20px;'>We prove a rigidity result for cocycles from higher rank lattices to <inline-formula><tex-math id="M2">\begin{document}$ \mathrm{Out}(F_N) $\end{document}</tex-math></inline-formula> and more generally to the outer automorphism group of a torsion-free hyperbolic group. More precisely, let <inline-formula><tex-math id="M3">\begin{document}$ G $\end{document}</tex-math></inline-formula> be either a product of connected higher rank simple algebraic groups over local fields, or a lattice in such a product. Let <inline-formula><tex-math id="M4">\begin{document}$ G \curvearrowright X $\end{document}</tex-math></inline-formula> be an ergodic measure-preserving action on a standard probability space, and let <inline-formula><tex-math id="M5">\begin{document}$ H $\end{document}</tex-math></inline-formula> be a torsion-free hyperbolic group. We prove that every Borel cocycle <inline-formula><tex-math id="M6">\begin{document}$ G\times X\to \mathrm{Out}(H) $\end{document}</tex-math></inline-formula> is cohomologous to a cocycle with values in a finite subgroup of <inline-formula><tex-math id="M7">\begin{document}$ \mathrm{Out}(H) $\end{document}</tex-math></inline-formula>. This provides a dynamical version of theorems of Farb–Kaimanovich–Masur and Bridson–Wade asserting that every homomorphism from <inline-formula><tex-math id="M8">\begin{document}$ G $\end{document}</tex-math></inline-formula> to either the mapping class group of a finite-type surface or the outer automorphism group of a free group, has finite image.</p><p style='text-indent:20px;'>The main new geometric tool is a barycenter map that associates to every triple of points in the boundary of the (relative) free factor graph a finite set of (relative) free splittings.</p>

Pointwise ergodic theorem for locally countable quasi-pmp graphs

We prove a pointwise ergodic theorem for quasi-probability-measure-preserving (quasi-pmp) locally countable measurable graphs, equivalently, Schreier graphs of quasi-pmp actions of countable groups. For ergodic graphs, the theorem gives an increasing sequence of Borel subgraphs with finite connected components over which the averages of any $ L^1 $ function converges to its expectation. This implies that every (not necessarily pmp) locally countable ergodic Borel graph on a standard probability space contains an ergodic hyperfinite subgraph. A consequence of this is that every ergodic treeable equivalence relation has an ergodic hyperfinite free factor.The pmp case of the main theorem was first proven by R. Tucker-Drob using a deep result from probability theory. Our proof is different: it is self-contained and applies more generally to quasi-pmp graphs. Among other things, it involves introducing a graph invariant concerning asymptotic averages of functions and a method of tiling a large part of the space with finite sets with prescribed properties. The non-pmp setting additionally exploits a new quasi-order called visibility to analyze the interplay between the Radon–Nikodym cocycle and the graph structure, providing a sufficient condition for hyperfiniteness.

On the bounded cohomology for ergodic nonsingular actions of amenable groups

Let Γ be an amenable countable discrete group. Fix an ergodic free non-singular action of Γ on a nonatomic standard probability space. Let G be a compactly generated locally compact second countable group such that the closure of the group of inner automorphisms of G is compact in the natural topology. It is shown that there exists a bounded ergodic G-valued cocycle of Γ.

Groups with infinite FC-center have the Schmidt property

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.

A theorem of Besicovitch and a generalization of the Birkhoff Ergodic Theorem

A remarkable theorem of Besicovitch is that an integrable function f f on R 2 \mathbb {R}^2 is strongly differentiable if its associated strong maximal function M S f M_S f is finite a.e. We provide an analogue of Besicovitch’s result in the context of ergodic theory that provides a generalization of Birkhoff’s Ergodic Theorem. In particular, we show that if f f is a measurable function on a standard probability space and T T is an invertible measure-preserving transformation on that space, then the ergodic averages of f f with respect to T T converge a.e. if and only if the associated ergodic maximal function T ∗ f T^*f is finite a.e.

A Unified Approach to Structural Limits and Limits of Graphs with Bounded Tree-Depth

In this paper we introduce a general framework for the study of limits of relational structures in general and graphs in particular, which is based on a combination of model theory and (functional) analysis. We show how the various approaches to graph limits fit to this framework and that they naturally appear as "tractable cases" of a general theory. As an outcome of this, we provide extensions of known results. We believe that this put these into next context and perspective. For example, we prove that the sparse--dense dichotomy exactly corresponds to random free graphons. The second part of the paper is devoted to the study of sparse structures. First, we consider limits of structures with bounded diameter connected components and we prove that in this case the convergence can be "almost" studied component-wise. We also propose the structure of limits objects for convergent sequences of sparse structures. Eventually, we consider the specific case of limits of colored rooted trees with bounded height and of graphs with bounded tree-depth, motivated by their role of elementary brick these graphs play in decompositions of sparse graphs, and give an explicit construction of a limit object in this case. This limit object is a graph built on a standard probability space with the property that every first-order definable set of tuples is measurable. This is an example of the general concept of {\em modeling} we introduce here. Our example is also the first "intermediate class" with explicitly defined limit structures.

Compact actions whose orbit equivalence relations are not profinite

Let Γ↷(X,μ) be a measure preserving action of a countable group Γ on a standard probability space (X,μ). We prove that if the action Γ↷X is not profinite and satisfies a certain spectral gap condition, then there does not exist a countable-to-one Borel homomorphism from its orbit equivalence relation to the orbit equivalence relation of any modular action (i.e., an inverse limit of actions on countable sets). As a consequence, we show that if Γ is a countable dense subgroup of a compact non-profinite group G such that the left translation action Γ↷G has spectral gap, then Γ↷G is antimodular and not orbit equivalent to any, not necessarily free, profinite action. This provides the first such examples of compact actions, partially answering a question of Kechris and answering a question of Tsankov.

Multiplication of weak equivalence classes may be discontinuous

For a countably infinite group Γ \Gamma , let W Γ {\mathcal {W}}_{\Gamma } denote the space of all weak equivalence classes of measure-preserving actions of Γ {\Gamma } on atomless standard probability spaces, equipped with the compact metrizable topology introduced by Abért and Elek. There is a natural multiplication operation on W Γ {\mathcal {W}}_{\Gamma } (induced by taking products of actions) that makes W Γ {\mathcal {W}}_{\Gamma } an Abelian semigroup. Burton, Kechris, and Tamuz showed that if Γ {\Gamma } is amenable, then W Γ {\mathcal {W}}_{\Gamma } is a topological semigroup; i.e., the product map W Γ × W Γ → W Γ : ( a , b ) ↦ a × b {\mathcal {W}}_{\Gamma } \times {\mathcal {W}}_{\Gamma } \to {\mathcal {W}}_{\Gamma } \colon (\mathfrak {a}, \mathfrak {b}) \mapsto \mathfrak {a} \times \mathfrak {b} is continuous. In contrast to that, we prove that if Γ {\Gamma } is a Zariski dense subgroup of S L d ( Z ) {\mathrm {SL}}_d({\mathbb {Z}}) for some d ⩾ 2 d \geqslant 2 (for instance, if Γ {\Gamma } is a non-Abelian free group), then multiplication on W Γ {\mathcal {W}}_{\Gamma } is discontinuous, even when restricted to the subspace F W Γ {\mathcal {FW}}_{\Gamma } of all free weak equivalence classes.

Weak containment of measure-preserving group actions

This paper concerns the study of the global structure of measure-preserving actions of countable groups on standard probability spaces. Weak containment is a hierarchical notion of complexity of such actions, motivated by an analogous concept in the theory of unitary representations. This concept gives rise to an associated notion of equivalence of actions, called weak equivalence, which is much coarser than the notion of isomorphism (conjugacy). It is well understood now that, in general, isomorphism is a very complex notion, a fact which manifests itself, for example, in the lack of any reasonable structure in the space of actions modulo isomorphism. On the other hand, the space of weak equivalence classes is quite well behaved. Another interesting fact that relates to the study of weak containment is that many important parameters associated with actions, such as the type, cost, and combinatorial parameters, turn out to be invariants of weak equivalence and in fact exhibit desirable monotonicity properties with respect to the pre-order of weak containment, a fact that can be useful in certain applications. There has been quite a lot of activity in this area in the last few years, and our goal in this paper is to provide a survey of this work.

EXISTENCE OF MODELING LIMITS FOR SEQUENCES OF SPARSE STRUCTURES

AbstractA sequence of graphs is FO-convergent if the probability of satisfaction of every first-order formula converges. A graph modeling is a graph, whose domain is a standard probability space, with the property that every definable set is Borel. It was known that FO-convergent sequence of graphs do not always admit a modeling limit, but it was conjectured that FO-convergent sequences of sufficiently sparse graphs have a modeling limits. Precisely, two conjectures were proposed:1.If a FO-convergent sequence of graphs is residual, that is if for every integer d the maximum relative size of a ball of radius d in the graphs of the sequence tends to zero, then the sequence has a modeling limit.2.A monotone class of graphs ${\cal C}$ has the property that every FO-convergent sequence of graphs from ${\cal C}$ has a modeling limit if and only if ${\cal C}$ is nowhere dense, that is if and only if for each integer p there is $N\left( p \right)$ such that no graph in ${\cal C}$ contains the pth subdivision of a complete graph on $N\left( p \right)$ vertices as a subgraph.In this article we prove both conjectures. This solves some of the main problems in the area and among others provides an analytic characterization of the nowhere dense–somewhere dense dichotomy.

Measurably entire functions and their growth

In 1997 B. Weiss introduced the notion of measurably entire functions and proved that they exist on every arbitrary free C-action defined on a standard probability space. In the same paper he asked about the minimal possible growth rate of such functions. In this work we show that for every arbitrary free C-action defined on a standard probability space there exists a measurably entire function whose growth rate does not exceed exp(exp[logp |z|]) for any p > 3. This complements a recent result by Buhovski, Glucksam, Logunov and Sodin who showed that such functions cannot have a growth rate smaller than exp(exp[logp |z|]) for any p < 2.

Measure-preserving semiflows and one-parameter Koopman semigroups

For a finite measure space $$\mathrm {X}$$ , we characterize strongly continuous Markov lattice semigroups on $$\mathrm {L}^p(\mathrm {X})$$ by showing that their generator A acts as a derivation on the dense subspace $$D(A)\cap \mathrm {L}^\infty (\mathrm {X})$$ . We then use this to characterize Koopman semigroups on $$\mathrm {L}^p(\mathrm {X})$$ if $$\mathrm {X}$$ is a standard probability space. In addition, we show that every measurable and measure-preserving flow on a standard probability space is isomorphic to a continuous flow on a compact Borel probability space.

Two special subgroups of the universal sofic group

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.

Measure concentration and the weak Pinsker property

Let $(X,\mu)$ be a standard probability space. An automorphism $T$ of $(X,\mu)$ has the weak Pinsker property if for every $\varepsilon > 0$ it has a splitting into a direct product of a Bernoulli shift and an automorphism of entropy less than $\varepsilon$. This property was introduced by Thouvenot, who asked whether it holds for all ergodic automorphisms. This paper proves that it does. The proof actually gives a more general result. Firstly, it gives a relative version: any factor map from one ergodic automorphism to another can be enlarged by arbitrarily little entropy to become relatively Bernoulli. Secondly, using some facts about relative orbit equivalence, the analogous result holds for all free and ergodic measure-preserving actions of a countable amenable group. The key to this work is a new result about measure concentration. Suppose now that $\mu$ is a probability measure on a finite product space $A^n$, and endow this space with its Hamming metric. We prove that $\mu$ may be represented as a mixture of other measures in which (i) most of the weight in the mixture is on measures that exhibit a strong kind of concentration, and (ii) the number of summands is bounded in terms of the difference between the Shannon entropy of $\mu$ and the combined Shannon entropies of its marginals.

On the classification problem of free ergodic actions of nonamenable groups

We show that, for any countable discrete nonamenable group Γ, the relations of conjugacy, orbit equivalence, stable orbit equivalence, von Neumann equivalence, and stable von Neumann equivalence of free ergodic pmp actions of Γ on the standard atomless probability space are not Borel. This answers a question of Kechris.

Some new examples of universal hypercyclic operators in the sense of Glasner and Weiss

— A bounded operator on a real or complex separable infinite-dimensional Banach space Z is universal in the sense of Glasner and Weiss if for every invertible ergodic measure-preserving transformation T of a standard Lebesgue probability space (X,B, μ), there exists an A-invariant probability measure ν on H with full support such that the two dynamical systems (X,B, μ;T ) and (H,BH , ν;A) are isomorphic. We present a general and simple criterion for an operator to be universal, which allows us to characterize universal operators among unilateral or bilateral weighted shifts on `p or c0, show the existence of universal operators on a large class of Banach spaces, and give a criterion for universality in terms of unimodular eigenvectors. We also obtain similar results for operators which are universal for all ergodic systems (not only for invertible ones), and study necessary conditions for an operator on a Hilbert space to be universal.