The empirical process of a short-range dependent stationary sequence under Gaussian subordination

Summary Consider the stationary sequenceX ₁=G(Z ₁),X ₂=G(Z ₂),..., whereG(·) is an arbitrary Borel function andZ ₁,Z ₂,... is a mean-zero stationary Gaussian sequence with covariance functionr(k)=E(Z ₁ Z _k+1) satisfyingr(0)=1 and _k=1 |r(k)| ^m < , where, withI{·} denoting the indicator function andF(·) the continuous marginal distribution function of the sequence {X _n }, the integerm is the Hermite rank of the family {I{G(·) x} –F(x):xR}. LetF _n (·) be the empirical distribution function ofX ₁,...,X _n . We prove that, asn, the empirical processn ^1/2{F _n (·)-F(·)} converges in distribution to a Gaussian process in the spaceD[–,].Partially supported by NSF Grant DMS-9208067     

A generalization of the Strassen converse

This note is devoted to a generalization of the Strassen converse. Let gn:R^∞→[0,∞], n?1 be a sequence of measurable functions such that, for every n?1, and for all x,y∈R^∞, where 0<C<∞ is a constant which is independent of n. Let be a sequence of i.i.d. random variables. Assume that there exist r?1 and a function ?:[0,∞)→[0,∞) with limt→∞?(t)=∞, depending only on the sequence such that lim supn→∞gn(X₁,X₂,…)=?(Er|X|) a.s. whenever Er|X|<∞ and EX=0. We prove the converse result, namely that lim supn→∞gn(X₁,X₂,…)<∞ a.s. implies Er|X|<∞ (and EX=0 if, in addition, lim supn→∞gn(c,c,…)=∞ for all c≠0). Some applications are provided to illustrate this result.     

o-Boundedness of free topological groups

Assuming the absence of Q-points (which is consistent with ZFC) we prove that the free topological group F(X) over a Tychonov space X is o-bounded if and only if every continuous metrizable image T of X satisfies the selection principle fin_?(O,Ω) (the latter means that for every sequence 〈un〉_n∈ω of open covers of T there exists a sequence 〈vn〉_n∈ω such that vn∈[un]^<ω and for every F∈[X]^<ω there exists n∈ω with F⊂?vn). This characterization gives a consistent answer to a problem posed by C. Hernándes, D. Robbie, and M. Tkachenko in 2000.     

Pointwise characterizations of Besov and Triebel–Lizorkin spaces and quasiconformal mappings

In this paper, the authors characterize, in terms of pointwise inequalities, the classical Besov spaces and Triebel–Lizorkin spaces for all s∈(0,1) and p,q∈(n/(n+s),∞], both in Rn and in the metric measure spaces enjoying the doubling and reverse doubling properties. Applying this characterization, the authors prove that quasiconformal mappings preserve on Rn for all s∈(0,1) and q∈(n/(n+s),∞]. A metric measure space version of the above morphism property is also established.     

Equilibrium states for factor maps between subshifts

Let π:X→Y be a factor map, where (X,σX) and (Y,σY) are subshifts over finite alphabets. Assume that X satisfies weak specification. Let a=(a₁,a₂)∈R² with a₁>0 and a₂?0. Let f be a continuous function on X with sufficient regularity (Hölder continuity, for instance). We show that there is a unique shift invariant measure μ on X that maximizes . In particular, taking f≡0 we see that there is a unique invariant measure μ on X that maximizes the weighted entropy a₁hμ(σX)+a₂hμ°π−1(σY), which answers an open question raised by Gatzouras and Peres (1996) in [15]. An extension is given to high dimensional cases. As an application, we show that for each compact invariant set K on the k-torus under a diagonal endomorphism, if the symbolic coding of K satisfies weak specification, then there is a unique invariant measure μ supported on K so that dimHμ=dimHK.     

The index formula and the spectral shift function for relatively trace class perturbations

We compute the Fredholm index, index(DA), of the operator DA=(d/dt)+A on L²(R;H) associated with the operator path , where (Af)(t)=A(t)f(t) for a.e. t∈R, and appropriate f∈L²(R;H), via the spectral shift function ξ(⋅;A₊,A₋) associated with the pair (A₊,A₋) of asymptotic operators A_±=A(±∞) on the separable complex Hilbert space H in the case when A(t) is generally an unbounded (relatively trace class) perturbation of the unbounded self-adjoint operator A₋.We derive a formula (an extension of a formula due to Pushnitski) relating the spectral shift function ξ(⋅;A₊,A₋) for the pair (A₊,A₋), and the corresponding spectral shift function ξ(⋅;H₂,H₁) for the pair of operators in this relative trace class context,This formula is then used to identify the Fredholm index of DA with ξ(0;A₊,A₋). In addition, we prove that index(DA) coincides with the spectral flow of the family {A(t)}_t∈R and also relate it to the (Fredholm) perturbation determinant for the pair (A₊,A₋): with the choice of the branch of ln(detH(⋅)) on C₊ such thatWe also provide some applications in the context of supersymmetric quantum mechanics to zeta function and heat kernel regularized spectral asymmetries and the eta-invariant.     

BLO spaces associated with the Ornstein–Uhlenbeck operator

Let (Rn,|⋅|,dγ) be the Gauss measure metric space, where Rn denotes the n-dimensional Euclidean space, |⋅| the Euclidean norm and for all x∈Rn the Gauss measure. In this paper, for any a∈(0,∞), the authors introduce some BLOa(γ) space, namely, the space of functions with bounded lower oscillation associated with a given class of admissible balls with parameter a. Then the authors prove that the noncentered local natural Hardy–Littlewood maximal operator is bounded from BMO(γ) of Mauceri and Meda to BLOa(γ). Moreover, a characterization of the space BLOa(γ), via the local natural maximal operator and BMO(γ), is given. The authors further prove that a class of maximal singular integrals, including the corresponding maximal operators of both imaginary powers of the Ornstein–Uhlenbeck operator and Riesz transforms of any order associated with the Ornstein–Uhlenbeck operator, are bounded from L^∞(γ) to BLOa(γ).     

Ideal convergence of continuous functions

For a given ideal I⊆P(ω), IC(I) denotes the class of separable metric spaces X such that whenever is a sequence of continuous functions convergent to zero with respect to the ideal I then there exists a set of integers {m₀<m₁<?} from the dual filter F(I) such that limi→∞fmi(x)=0 for all x∈X. We prove that for the most interesting ideals I, IC(I) contains only singular spaces. For example, if I=Id is the asymptotic density zero ideal, all IC(Id) spaces are perfectly meager while if I=Ib is the bounded ideal then IC(Ib) spaces are σ-sets.     

Continuous extension operators and convexity

Given a nonempty closed subset A of a Hilbert space X, we denote by L(A) the space of all bounded Lipschitz mappings from A into X, equipped with the supremum norm. We show that there is a continuous mapping F_c:L(A)?L(X) such that for each g∈L(A), F_c(g)|_A=g, , and . We also prove that the corresponding set-valued extension operator is lower semicontinuous.     

Characterizing filters by convergence (with respect to filters) in Banach spaces

Let X be a topological space and let F be a filter on N, recall that a sequence (xn)_n∈N in X is said to be F-convergent to the point x∈X, if for each neighborhood U of x, {n∈N:xn∈U}∈F. By using F-convergence in ?₁ and in Banach spaces, we characterize the P-filters, the P-filters⁺, the weak P-filters, the Q-filters, the Q-filters⁺, the weak Q-filters, the selective filters and the selective⁺ filters.     

On extensions of countable filterbases to ultrafilters and ultrafilter compactness

We show in the Zermelo-Fraenkel set theory ZF without the axiom of choice:

1. Given an infinite set X, the Stone space S(X) is ultrafilter compact.

2. For every infinite set X, every countable filterbase of X extends to an ultra-filter i? for every infinite set X, S(X) is countably compact.

3. ω has a free ultrafilter i? every countable, ultrafilter compact space is countably compact.

   We also show the following:

4. There are a permutation model 𝒩 and a set X ∈ 𝒩 such that X has no free ultrafilters and S(X) is not compact but S(X) is countably compact and every countable filterbase of X extends to an ultrafilter.

5. It is relatively consistent with ZF that every countable filterbase of ω extends to an ultrafilter but there exists a countable filterbase of ? which does not extend to an ultrafilter. Hence, it is relatively consistent with ZF that ? has free ultrafilters but there exists a countable filterbase of ? which does not extend to an ultrafilter.

     

Characterizing omega-limit sets which are closed orbits

Let X be a vector field in a compact n-manifold M, n?2. Given Σ⊂M we say that q∈M satisfies (P)_Σ if the closure of the positive orbit of X through q does not intersect Σ, but, however, there is an open interval I with q as a boundary point such that every positive orbit through I intersects Σ. Among those q having saddle-type hyperbolic omega-limit set ω(q) the ones with ω(q) being a closed orbit satisfy (P)_Σ for some closed subset Σ. The converse is true for n=2 but not for n?4. Here we prove the converse for n=3. Moreover, we prove for n=3 that if ω(q) is a singular-hyperbolic set [C. Morales, M. Pacifico, E. Pujals, On C¹ robust singular transitive sets for three-dimensional flows, C. R. Acad. Sci. Paris Sér. I 26 (1998) 81-86], [C. Morales, M. Pacifico, E. Pujals, Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers, Ann. of Math. (2) 160 (2) (2004) 375-432], then ω(q) is a closed orbit if and only if q satisfies (P)_Σ for some Σ closed. This result improves [S. Bautista, Sobre conjuntos hiperbólicos-singulares (On singular-hyperbolic sets), thesis Uiversidade Federal do Rio de Janeiro, 2005 (in Portuguese)] and [C. Morales, M. Pacifico, Mixing attractors for 3-flows, Nonlinearity 14 (2001) 359-378].     

On holomorphic domination, I

Let X be a separable Banach space and u:X→R locally upper bounded. We show that there are a Banach space Z and a holomorphic function h:X→Z with u(x)<‖h(x)‖ for x∈X. As a consequence we find that the sheaf cohomology group Hq(X,O) vanishes if X has the bounded approximation property (i.e., X is a direct summand of a Banach space with a Schauder basis), O is the sheaf of germs of holomorphic functions on X, and q?1. As another consequence we prove that if f is a C¹-smooth -closed (0,1)-form on the space X=L₁[0,1] of summable functions, then there is a C¹-smooth function u on X with on X.     

On the Laczkovich-Komjáth property of sigma-ideals

Komjáth in 1984 proved that, for each sequence (An) of analytic subsets of a Polish space X, if lim supn∈HAn is uncountable for every H∈ω[N] then _?n∈GAn is uncountable for some G∈ω[N]. This fact, by our definition, means that the σ-ideal [X]^?ω has property (LK). We prove that every σ-ideal generated by X/E has property (LK), for an equivalence relation E⊂X² of type Fσ with uncountably many equivalence classes. We also show the parametric version of this result. Finally, the invariance of property (LK) with respect to various operations is studied.     

Extracting information is hard: A Turing degree of non-integral effective Hausdorff dimension

We construct a infinite binary sequence with effective Hausdorff dimension 1/2 that does not compute a sequence of higher dimension. Introduced by Lutz, effective Hausdorff dimension can be viewed as a measure of the information density of a sequence. In particular, the dimension of A∈ω² is the lim inf of the ratio between the information content and length of initial segments of A. Thus the main result demonstrates that it is not always possible to extract information from a partially random source to produce a sequence that has higher information density.     

Extreme points of Banach lattices related to conditional expectations

Let (X,F,μ) be a complete probability space, B a sub-σ-algebra, and Φ the probabilistic conditional expectation operator determined by B. Let K be the Banach lattice {f∈L¹(X,F,μ):‖Φ(|f|)_‖∞<∞} with the norm ‖f‖=‖Φ(|f|)_‖∞. We prove the following theorems:

(1)

The closed unit ball of K contains an extreme point if and only if there is a localizing set E for B such that supp(Φ(χE))=X.

(2)

Suppose that there is n∈N such that f?nΦ(f) for all positive f in L^∞(X,F,μ). Then K has the uniformly λ-property and every element f in the complex K with is a convex combination of at most 2n extreme points in the closed unit ball of K.

     

Asymptotic cone of semisimple orbits for symmetric pairs

Let G be a reductive algebraic group over C and denote its Lie algebra by g. Let Oh be a closed G-orbit through a semisimple element h∈g. By a result of Borho and Kraft (1979) [4], it is known that the asymptotic cone of the orbit Oh is the closure of a Richardson nilpotent orbit corresponding to a parabolic subgroup whose Levi component is the centralizer ZG(h) in G. In this paper, we prove an analogue on a semisimple orbit for a symmetric pair.More precisely, let θ be an involution of G, and K=Gθ a fixed point subgroup of θ. Then we have a Cartan decomposition g=k+s of the Lie algebra g=Lie(G) which is the eigenspace decomposition of θ on g. Let {x,h,y} be a normal sl₂ triple, where x,y∈s are nilpotent, and h∈k semisimple. In addition, we assume , where denotes the complex conjugation which commutes with θ. Then is a semisimple element in s, and we can consider a semisimple orbit Ad(K)a in s, which is closed. Our main result asserts that the asymptotic cone of Ad(K)a in s coincides with , if x is even nilpotent.     

The endomorphism spectrum of an ordered set

Theendomorphism spectrum of an ordered setP, spec(P)={|f(P)|:f End(P)} andspectrum number, sp(P)=max(spec(P)\{|P|}) are introduced. It is shown that |P|>(1/2)n(n – 1) ^{n – 1} implies spec(P) = {1, 2, ...,n} and that if a projective plane of ordern exists, then there is an ordered setP of size 2n ²+2n+2 with spec(P)={1, 2, ..., 2n+2, 2n+4}. Lettingh(n)=max{|P|: sp(P)n}, it follows thatc ₁ n ²h(n)c ₂ n ⁿ⁺¹ for somec ₁ andc ₂. The lower bound disproves the conjecture thath(n)2n. It is shown that if |P| – 1 spec(P) thenP has a retract of size |P| – 1 but that for all there is a bipartite ordered set with spec(P) = {|P| – 2, |P| – 4, ...} which has no proper retract of size|P| – . The case of reflexive graphs is also treated.Partially supported by a grant from the NSERC.Partially supported by a grant from the NSERC.