Probability Measureμ Research Articles

Definable Kőnig theorems

Let X X be a Polish space with Borel probability measure μ , \mu , and let G G be a Borel graph on X X with no odd cycles and maximum degree Δ ( G ) . \Delta (G). We show that the Baire measurable edge chromatic number of G G is at most Δ ( G ) + 1 \Delta (G)+1 , and if G G is μ \mu -hyperfinite then the μ \mu -measurable edge chromatic number obeys the same bound. More generally, we show that G G has Borel edge chromatic number at most Δ ( G ) \Delta (G) plus its asymptotic separation index.

Orthogonality relations on certain homogeneous spaces

Let G G be a locally compact group and let K K be its closed subgroup. Write G ^ K \widehat {G}_{K} for the set of irreducible representations with non-zero K K -invariant vectors. We call a pair ( G , K ) (G,K) admissible if for each irreducible representation ( π , V π ) (\pi , V_{\pi }) in G ^ K \widehat {G}_{K} , its K K -invariant subspace V π K V_{\pi }^{K} is of finite dimension. For each π \pi in G ^ K \widehat {G}_{K} , let π v i , ξ ¯ j \pi _{v_{i}, \overline {\xi }_{j}} ’s ( π v i , ξ ¯ j ( g K ) ≔ ⟨ v i , π ( g ) ξ j ⟩ ) (\pi _{v_{i}, \overline {\xi }_{j}}(gK)≔\langle v_{i}, \pi (g)\xi _{j}\rangle ) be the matrix coefficeints on G / K G/K induced by fixed orthonormal bases { v i } \{v_{i}\} and { ξ j } \{\xi _{j}\} for V π V_{\pi } and V π K V_{\pi }^{K} respectively. A probability measure μ \mu on G / K G/K is called a spectral measure if there is a subset Γ \Gamma of G ^ K \widehat {G}_{K} such that the set of all such matrix coefficients π v i , ξ ¯ j , π ∈ Γ , \pi _{v_{i}, \overline {\xi }_{j}},\ \pi \in \Gamma , constitutes an orthonormal basis for L 2 ( G / K , μ ) L^{2}(G/K, \mu ) with some suitable normalization of these matrix coordinate functions. In this paper, we shall give a characterization of a spectral measure for an admissible pair ( G , K ) (G,K) by using the Fourier transform on G / K G/K . Also, from this we show that there is a “local translation” (we call it locally regular representation in the sequel) of G G on L 2 ( G / K , μ ) L^{2}(G/K, \mu ) under a mild condition. This will give us some necessary conditions for the existence of spectral measures. In particular, the atomic spectral measures of finite supports for Gelfand pairs are studied.

On the measure of maximal entropy for finite horizon Sinai Billiard maps

The Sinai billiard map T T on the two-torus, i.e., the periodic Lorentz gas, is a discontinuous map. Assuming finite horizon, we propose a definition h ∗ h_* for the topological entropy of T T . We prove that h ∗ h_* is not smaller than the value given by the variational principle, and that it is equal to the definitions of Bowen using spanning or separating sets. Under a mild condition of sparse recurrence to the singularities, we get more: First, using a transfer operator acting on a space of anisotropic distributions, we construct an invariant probability measure μ ∗ \mu _* of maximal entropy for T T (i.e., h μ ∗ ( T ) = h ∗ h_{\mu _*}(T)=h_* ), we show that μ ∗ \mu _* has full support and is Bernoulli, and we prove that μ ∗ \mu _* is the unique measure of maximal entropy and that it is different from the smooth invariant measure except if all nongrazing periodic orbits have multiplier equal to h ∗ h_* . Second, h ∗ h_* is equal to the Bowen–Pesin–Pitskel topological entropy of the restriction of T T to a noncompact domain of continuity. Last, applying results of Lima and Matheus, as upgraded by Buzzi, the map T T has at least C e n h ∗ C e^{nh_*} periodic points of period n n for all n ∈ N n \in \mathbb {N} .

On logarithmically Benford Sequences

Let I ⊂ N \mathcal {I} \subset \mathbb {N} be an infinite subset, and let { a i } i ∈ I \{a_i\}_{i \in \mathcal {I}} be a sequence of nonzero real numbers indexed by I \mathcal {I} such that there exist positive constants m , C 1 m, C_1 for which | a i | ≤ C 1 ⋅ i m |a_i| \leq C_1 \cdot i^m for all i ∈ I i \in \mathcal {I} . Furthermore, let c i ∈ [ − 1 , 1 ] c_i \in [-1,1] be defined by c i = a i C 1 ⋅ i m c_i = \frac {a_i}{C_1 \cdot i^m} for each i ∈ I i \in \mathcal {I} , and suppose the c i c_i ’s are equidistributed in [ − 1 , 1 ] [-1,1] with respect to a continuous, symmetric probability measure μ \mu . In this paper, we show that if I ⊂ N \mathcal {I} \subset \mathbb {N} is not too sparse, then the sequence { a i } i ∈ I \{a_i\}_{i \in \mathcal {I}} fails to obey Benford’s Law with respect to arithmetic density in any sufficiently large base, and in fact in any base when μ ( [ 0 , t ] ) \mu ([0,t]) is a strictly convex function of t ∈ ( 0 , 1 ) t \in (0,1) . Nonetheless, we also provide conditions on the density of I ⊂ N \mathcal {I} \subset \mathbb {N} under which the sequence { a i } i ∈ I \{a_i\}_{i \in \mathcal {I}} satisfies Benford’s Law with respect to logarithmic density in every base. As an application, we apply our general result to study Benford’s Law-type behavior in the leading digits of Frobenius traces of newforms of positive, even weight. Our methods of proof build on the work of Jameson, Thorner, and Ye, who studied the particular case of newforms without complex multiplication.

Hausdorff measures, dimensions and mutual singularity

Let ( X , d ) (X,d) be a metric space. For a probability measure μ \mu on a subset E E of X X and a Vitali cover V V of E E , we introduce the notion of a b μ b_{\mu } -Vitali subcover V μ V_{\mu } , and compare the Hausdorff measures of E E with respect to these two collections. As an application, we consider graph directed self-similar measures μ \mu and ν \nu in R d \mathbb {R}^{d} satisfying the open set condition. Using the notion of pointwise local dimension of μ \mu with respect to ν \nu , we show how the Hausdorff dimension of some general multifractal sets may be computed using an appropriate stochastic process. As another application, we show that Olsen’s multifractal Hausdorff measures are mutually singular.

On furstenberg’s characterization of harmonic functions on symmetric spaces

We provide a necessary and sufficient condition on a radial probability measureμ on a symmetric space for whichf =f *μ, f bounded, implies thatf is harmonic. In particular, we obtain a short and elementary proof of a theorem of Furstenberg which says that iff is a bounded function on a symmetric space which satisfiesf =f *μ for some radialabsolutely continuous probability measureμ, thenf is harmonic.

Ergodicity of conformal measures for unimodal polynomials

Let f f be a polynomial and μ \mu a conformal measure for f f , i.e., a Borel probability measure μ \mu with Jacobian equal to | D f ( z ) | δ |Df(z)|^{\delta } . We show that if f f is a real unimodal polynomial (a polynomial with just one critical point), then μ \mu is ergodic. We also show that μ \mu is ergodic if f f is a complex unimodal polynomial with one parabolic periodic point or a quadratic polynomial in the S L \mathcal {SL} class with a priori bounds (as defined in Lyubich (1997)).

A simple characterization of the set ofμ-entropy pairs and applications

We present simple characterizations of the setsE μ andE X of measure entropy pairs and topological entropy pairs of a topological dynamical system (X, T) with invariant probability measureμ. This characterization is used to show that the set of (measure) entropy pairs of a product system coincides with the product of the sets of (measure) entropy pairs of the component systems; in particular it follows that the product of u.p.e. systems (topological K-systems) is also u.p.e. Another application is to show that the proximal relationP forms a residual subset of the setE X . Finally an example of a minimal point distal dynamical system is constructed for whichE X ∩(X 0×X 0)≠ $$\not 0$$ , whereX 0 is the denseG δ subset of distal points inX.

Collapsible probability measures and concentration functions on Lie groups

Given a locally compact group G and a probability measure μ on G it is of interest to know, in various situations, whether there exist divergent sequences {gn} such that {gn μg−1n is relatively compact (see for example [DM3] and [DS]); this phenomenon may be viewed as ‘collapsing’ of the measure. It is the purpose of this note to prove Theorem 1 below and give certain applications to the asymptotic behaviour of concentration functions.

Subharmonicity without Upper Semicontinuity

LetΩbe an open subset ofRd(d⩾2). Givenx∈Ω, a Jensenmeasureforxis a Borel probability measureμ, supported on a compact subset ofΩ, such that each subharmonic functionuonΩsatisfiesu(x)⩽∫udμ. ((*))A functionuonΩ(subject to certain measurability conditions, much weaker for example than upper semicontinuity) is calledquasi-subharmonicif it satisfies (*) for eachx∈Ωand each Jensen measureμforx. Our first result is thatuis quasi-subharmonic onΩif and only if its upper semicontinuous regularizationu* is subharmonic onΩand equal tououtside a set of capacity zero. This easily implies, for example, Cartan's classical theorem on the supremum of a locally bounded family of subharmonic functions. Our second result is a converse to Cartan's theorem: provided thatΩhas a Green's function, every quasi-subharmonic function 4 onΩis the supremum of some family of subharmonic functions. These ideas eventually lead to the following duality theorem. Ifϕis a Borel function locally bounded above onΩ, and ifΩhas a Green's function, then for each x∈Ω,sup{v(x):vsubharmoniconΩv⩽ϕ}=inf∫ϕdμ:μisaJensenmeasureforx.We also investigate to what extent these results carry over to plurisubharmonic functions.

Convexity and Haar null sets

It is shown that for every closed, convex and nowhere dense subset C C of a superreflexive Banach space X X there exists a Radon probability measure μ \mu on X X so that μ ( C + x ) = 0 \mu (C+x)=0 for all x ∈ X x\in X . In particular, closed, convex, nowhere dense sets in separable superreflexive Banach spaces are Haar null. This is unlike the situation in separable nonreflexive Banach spaces, where there always exists a closed convex nowhere dense set which is not Haar null.

Random Approximants and Neural Networks

LetDbe a set with a probability measureμ,μ(D)=1, and letKbe a compact subset ofLq(D, μ), 1⩽q<∞. Forf∈Lq,n=1, 2, …, letρn(f, K)=inf‖f−gn‖q, where the infimum is taken over allgnof the formgn=∑ni=1aiφi, with arbitraryφi∈Kandai∈R. It is shown that for[formula], under some mild restrictions,ρn(f, K)⩽Cqεn(K)n−1/2, whereεn(K)→0 asn→∞. This fact is used to estimate the errors of certain neural net approximations. For the latter, also the lower estimates of errors are given.

On convolution powers on semidirect products

LetK be a compact group of linear operators of thed-dimensional spaceRd andGK,d denote the semidirect productK byRd. It is shown that if an adapted probability measureμ onGK,d is not scattered (i.e. for some compactF we havex0 ∈ Rd (gF)>0), then there exists a nonzero vectorx0 ∈Rd such thatk1(x0)=k2(x0) holds for all (k1,x1) and (k2,x2) belonging to the topological supportS(μ) of the measureμ. As a result we obtain that every adapted and strictly aperiodic probability measure on the group of all rigid motions of thed-dimensional Euclidian space is scattered.

Invariant measures for certain expansive ℤ2-actions

Letp>1 be prime, and letY ⊂X=(ℤ/pℤ)ℤ 2) be an infinite, closed, shift-invariant subgroup with the following properties: the restriction toY of the shift-actionσ of ℤ2 onX is mixing with respect to the Haar measureλ Y ofY, and every closed, shift-invariant subgroupZ ⊂Y is finite. We prove that every sufficiently mixing, non-atomic, shift-invariant probability measureμ onY is equal toλ Y .

A stochastic Hopf bifurcation

Let {x t :t≧0} be the solution of a stochastic differential equation (SDE) in ℝ d which fixes 0, and let λ denote the Lyapunov exponent for the linear SDE obtained by linearizing the original SDE at 0. It is known that, under appropriate conditions, the sign of λ controls the stability/instability of 0 and the transience/recurrence of {x t :t≧0} on ℝ d \{0}. In particular if the coefficients in the SDE depend on some parameterz which is varied in such a way that the corresponding Lyapunov exponentλ z changes sign from negative to positive the (almost-surely) stable fixed point at 0 is replaced by an (almost-surely) unstable fixed point at 0 together with an attracting invariant probability measureμ z on ℝ d \{0}. In this paper we investigate the limiting behavior ofμ z asλ z converges to 0 from above. The main result is that the rescaled measures (1/λ z )μ z converge (in an appropriate weak sense) to a non-trivial σ-finite measure on ℝ d \{0}.

A characterization of the exponential distribution involving absolute differences of i.i.d. random variables

A probability measure μ \mu on [ 0 , ∞ ) [0,\infty ) is said to have property H if for independent random variables X 1 {X_1} and X 2 {X_2} distributed according to μ \mu the absolute difference | X 1 − X 2 | |{X_1} - {X_2}| has the same distribution. Continuing previous work of Puri and Rubin we characterize the set of all distributions having property H.

Topological ergodic theory and mean rotation

Let H 0 ( M n , μ ) {\mathcal {H}_0}({M^n},\mu ) denote the set of all homeomorphisms of a compact manifold M n {M^n} that preserve a locally positive nonatomic Borel probability measure μ \mu and are isotopic to the identity. The notion of the mean rotation vector for a torus homeomorphism has been extended by Fathi to a continuous map θ \theta on H 0 ( M n , μ ) {\mathcal {H}_0}({M^n},\mu ) . We show that any abstract ergodic behavior typical for automorphisms of ( M n , μ ) ({M^n},\mu ) as a Lebesgue space is also typical not only in H 0 ( M n , μ {\mathcal {H}_0}({M^n},\mu but also in each closed subset of constant θ \theta . By typical we mean dense G δ {G_\delta } in the appropriate space. Weak mixing is an example of such a typical abstract ergodic behavior. This contrasts sharply with a deep result of the KAM theory that for some rotation vectors v → \overrightarrow v , there is an open neighborhood of rotation by v → \overrightarrow v , in the space of smooth volume preserving n n -torus diffeomorphisms with θ = v → \theta = \overrightarrow v , where each diffeomorphism in the open set is conjugate to rotation by v → \overrightarrow v (and hence cannot be weak mixing).

A multiplicative ergodic theorem for rotation numbers

Given a vector fieldX on a Riemannian manifoldM of dimension at least 2 whose flow leaves a probability measureμ invariant, the multiplicative ergodic theorem tells us thatμ-a.s. every tangent vector possesses a Lyapunov exponent (exponential growth rate) that is equal to one of finitely many basic exponents corresponding toX andμ. We prove that, in the case of a simple Lyapunov spectrum, every tangent planeμ-a.s. possesses a rotation number that is equal to one of finitely many basic rotation numbers corresponding toX andμ. Rotation in a plane is measured as the time average of the infinitesimal changes of the angle between a frame moved by the linearized flow and the same frame parallel-transported by a (canonical) connection.

Uniform algebras as Banach spaces

Let A be a closed subalgebra of the complex Banach algebra C(S), containing the constant functions. We assume that one has found a probability measureμ on S and a function F from L∞(μ) such that: 1)¦F¦= 1 a.e. relative to μ; 2) F μ e A1; 3) F is a limit point of the unit ball of the algebra A in the topology δ(L∞(μ), L1(μ)). One proves in the paper that under these conditions the space A** contains a complement space, isometric to H∞. The measure μ and the function F, satisfying the conditions l)-3) indeed exist if the maximal ideal space of the algebra A contains a non-one-point part (and it is very likely that such aμ. and F exist whenever the algebra A is not self-adjoint). Thus, the above-formulated result allows us to extend A. Pelczynski's theorem (Ref, Zh. Mat., 1975, 1B894) regarding the space H∞ to a very broad class of uniform algebras.

Vague convergence of sums of independent random variables

A sequence (μn) of probability measures on the real line is said to converge vaguely to a measureμ if∫ fdμn →∫ fdμ for every continuous functionf withcompact support. In this paper one investigates problems analogous to the classical central limit problem under vague convergence. Let ‖μ‖ denote the total mass ofμ andδ0 denote the probability measure concentrated in the origin. For the theory of infinitesimal triangular arrays it is true in the present context, as it is in the classical one, that all obtainable limit laws are limits of sequences of infinitely divisible probability laws. However, unlike the classical situation, the class of infinitely divisible laws is not closed under vague convergence. It is shown that for every probability measureμ there is a closed interval [0,λ], [0,e−1] ⊂ [0,λ] ⊂ [0, 1], such thatβμ is attainable as a limit of infinitely divisible probability laws iffβ e [0,λ]. In the independent identically distributed case, it is shown that if (x1 + ... +xn)/an, an → ∞, converges vaguely toμ with 0 x] is a slowly varying function ofx. Conversely, ifL(x) is slowly varying, then for everyβ e (0, 1) one can choosean → ∞ so that the limit measure=βδ0.