A Central Limit Theorem for Predictive Distributions

Let S be a Borel subset of a Polish space and F the set of bounded Borel functions f:S→R. Let an(·)=P(Xn+1∈·∣X1,…,Xn) be the n-th predictive distribution corresponding to a sequence (Xn) of S-valued random variables. If (Xn) is conditionally identically distributed, there is a random probability measure μ on S such that ∫fdan⟶a.s.∫fdμ for all f∈F. Define Dn(f)=dn∫fdan−∫fdμ for all f∈F, where dn>0 is a constant. In this note, it is shown that, under some conditions on (Xn) and with a suitable choice of dn, the finite dimensional distributions of the process Dn=Dn(f):f∈F stably converge to a Gaussian kernel with a known covariance structure. In addition, Eφ(Dn(f))∣X1,…,Xn converges in probability for all f∈F and φ∈Cb(R).

Packing topological entropy for amenable group actions

Packing topological entropy is a dynamical analogy of the packing dimension, which can be viewed as a counterpart of Bowen topological entropy. In the present paper we give a systematic study of the packing topological entropy for a continuous G-action dynamical system $(X,G)$ , where X is a compact metric space and G is a countable infinite discrete amenable group. We first prove a variational principle for amenable packing topological entropy: for any Borel subset Z of X, the packing topological entropy of Z equals the supremum of upper local entropy over all Borel probability measures for which the subset Z has full measure. Then we obtain an entropy inequality concerning amenable packing entropy. Finally, we show that the packing topological entropy of the set of generic points for any invariant Borel probability measure $\mu $ coincides with the metric entropy if either $\mu $ is ergodic or the system satisfies a kind of specification property.

Uniformly factoring weakly compact operators and parametrised dualisation

This article deals with the problem of when, given a collection $\mathcal {C}$ of weakly compact operators between separable Banach spaces, there exists a separable reflexive Banach space Z with a Schauder basis so that every element in $\mathcal {C}$ factors through Z (or through a subspace of Z). In particular, we show that there exists a reflexive space Z with a Schauder basis so that for each separable Banach space X, each weakly compact operator from X to $L_1[0,1]$ factors through Z. We also prove the following descriptive set theoretical result: Let $\mathcal {L}$ be the standard Borel space of bounded operators between separable Banach spaces. We show that if $\mathcal {B}$ is a Borel subset of weakly compact operators between Banach spaces with separable duals, then for $A \in \mathcal {B}$ , the assignment $A \to A^*$ can be realised by a Borel map $\mathcal {B}\to \mathcal {L}$ .

Dimension transformation formula for conformal maps into the complement of an SLE curve

We prove a formula relating the Hausdorff dimension of a deterministic Borel subset of $${\mathbb {R}}$$R and the Hausdorff dimension of its image under a conformal map from the upper half-plane to a complementary connected component of an $$\hbox {SLE}_\kappa $$SLEκ curve for $$\kappa \not =4$$κ≠4. Our proof is based on the relationship between SLE and Liouville quantum gravity together with the one-dimensional KPZ formula of Rhodes and Vargas (ESAIM Probab Stat 15:358–371, 2011) and the KPZ formula of Gwynne et al. (Ann Probab, 2015). As an intermediate step we prove a KPZ formula which relates the Euclidean dimension of a subset of an $$\hbox {SLE}_\kappa $$SLEκ curve for $$\kappa \in (0,4)\cup (4,8)$$κ∈(0,4)∪(4,8) and the dimension of the same set with respect to the $$\gamma $$γ-quantum natural parameterization of the curve induced by an independent Gaussian free field, $$\gamma = \sqrt{\kappa }\wedge (4/\sqrt{\kappa })$$γ=κ∧(4/κ).

Expansions of the Real Field by Canonical Products

We consider expansions of o-minimal structures on the real field by collections of restrictions to the positive real line of the canonical Weierstrass products associated with sequences such as $(-n^{s})_{n>0}$ (for $s>0$) and $(-s^{n})_{n>0}$ (for $s>1$), and also expansions by associated functions such as logarithmic derivatives. There are only three possible outcomes known so far: (i) the expansion is o-minimal (that is, definable sets have only finitely many connected components); (ii) every Borel subset of each $\mathbb{R}^{n}$ is definable; (iii) the expansion is interdefinable with a structure of the form $(\mathfrak{R}^{\prime },\unicode[STIX]{x1D6FC}^{\mathbb{Z}})$ where $\unicode[STIX]{x1D6FC}>1$, $\unicode[STIX]{x1D6FC}^{\mathbb{Z}}$ is the set of all integer powers of $\unicode[STIX]{x1D6FC}$, and $\mathfrak{R}^{\prime }$ is o-minimal and defines no irrational power functions.

Fubini Property for Microscopic Sets

In 2000, I. Recław and P. Zakrzewski introduced the notion of Fubini Property for the pair (I,J) of two σ-ideals in the following way. Let I and J be two σ-ideals on Polish spaces X and Y, respectively. The pair (I,J) has the Fubini Property (FP) if for every Borel subset B of X×Y such that all its vertical sections Bx = {y ∈ Y : (x, y) ∈ B} are in J, then the set of all y ∈ Y, for which horizontal section By = {x ∈ X : (x, y) ∈ B} does not belong to I, is a set from J, i.e., {y ∈ Y : By ∉ I} ∈ J. The Fubini property for the σ-ideal M of microscopic sets is considered and the proof that the pair (M,M) does not satisfy (FP) is given.

Sharp Inequalities for the Haar System and Fourier Multipliers

A classical result of Paley and Marcinkiewicz asserts that the Haar systemh=hkk≥0on0,1forms an unconditional basis ofLp0,1provided1<p<∞. That is, if𝒫Jdenotes the projection onto the subspace generated byhjj∈J(Jis an arbitrary subset ofℕ), then𝒫JLp0,1→Lp0,1≤βpfor some universal constantβpdepending only onp. The purpose of this paper is to study related restricted weak-type bounds for the projections𝒫J. Specifically, for any1≤p<∞we identify the best constantCpsuch that𝒫JχALp,∞0,1≤CpχALp0,1for everyJ⊆ℕand any Borel subsetAof0,1. In fact, we prove this result in the more general setting of continuous-time martingales. As an application, a related estimate for a large class of Fourier multipliers is established.

Ergodic measures of Markov semigroups with the e-property

We study the set of ergodic measures for a Markov semigroup on a Polish state space. The principal assumption on this semigroup is the e-property, an equicontinuity condition. We introduce a weak concentrating condition around a compact set K and show that this condition has several implications on the set of ergodic measures, one of them being the existence of a Borel subset K0 of K with a bijective map from K0 to the ergodic measures, by sending a point in K0 to the weak limit of the Cesàro averages of the Dirac measure on this point. We also give sufficient conditions for the set of ergodic measures to be countable and finite. Finally, we give a quite general condition under which the Cesàro averages of any measure converge to an invariant measure.

SOLUTION PROPERTIES OF SOME CLASSES OF OPERATOR EQUATIONS IN HILBERT SPACES

We study properties of solutions of the operator equation [Formula: see text], [Formula: see text], where [Formula: see text] a closable linear operator on a Hilbert space [Formula: see text], such that there exists a self-adjoint operator [Formula: see text] on [Formula: see text], with the resolution of identity E(·), which commutes with [Formula: see text]. We are interested in the question of regular admissibility of the subspace [Formula: see text], i.e. when for every [Formula: see text] there exists a unique (mild) solution u in [Formula: see text] of this equation. We introduce the notion of equation spectrum Σ associated with Eq. (*), and prove that if Λ ⊂ ℝ is a compact subset such that Λ ⋂ Σ = ∅, then [Formula: see text] is regularly admissible. If Λ ⊂ ℝ is an arbitrary Borel subset such that Λ ⋂ Σ = ∅, then, in general, [Formula: see text] needs not be regularly admissible, but we derive necessary and sufficient conditions, in terms of some inequalities, for the regular admissibility of [Formula: see text]. Our results are generalizations of the well-known spectral mapping theorem of Gearhart-Herbst-Howland-Prüss [4], [5], [6], [9], as well as of the recent results of Cioranescu-Lizama [3], Schüler [10] and Vu [11], [12].