Exact inequalities for sums of asymmetric random variables, with applications

Let be independent identically distributed random variables each having the standardized Bernoulli distribution with parameter . Let if and . Let . Let f be such a function that f and f′′ are nondecreasing and convex. Then it is proved that for all nonnegative numbers one has the inequality where . The lower bound on m is exact for each . Moreover, is Schur-concave in . A number of corollaries are obtained, including upper bounds on generalized moments and tail probabilities of (super)martingales with differences of bounded asymmetry, and also upper bounds on the maximal function of such (super)martingales. Applications to generalized self-normalized sums and t-statistics are given.      

Uniform central limit theorems for kernel density estimators

Let be the classical kernel density estimator based on a kernel K and n independent random vectors X _i each distributed according to an absolutely continuous law on . It is shown that the processes , , converge in law in the Banach space , for many interesting classes of functions or sets, some -Donsker, some just -pregaussian. The conditions allow for the classical bandwidths h _n that simultaneously ensure optimal rates of convergence of the kernel density estimator in mean integrated squared error, thus showing that, subject to some natural conditions, kernel density estimators are ‘plug-in’ estimators in the sense of Bickel and Ritov (Ann Statist 31:1033–1053, 2003). Some new results on the uniform central limit theorem for smoothed empirical processes, needed in the proofs, are also included.      

Inequalities of the Brunn–Minkowski type for Gaussian measures

Let be an integer, let γ be the standard Gaussian measure on , and let . Given this paper gives a necessary and sufficient condition such that the inequality is true for all Borel sets A ₁,...,A _m in of strictly positive γ-measure or all convex Borel sets A ₁,...,A _m in of strictly positive γ-measure, respectively. In particular, the paper exhibits inequalities of the Brunn–Minkowski type for γ which are true for all convex sets but not for all measurable sets.      

Hausdorff dimension of the contours of symmetric additive Lévy processes

Let X ₁, ..., X _N denote N independent, symmetric Lévy processes on R ^d . The corresponding additive Lévy process is defined as the following N-parameter random field on R ^d : Khoshnevisan and Xiao (Ann Probab 30(1):62–100, 2002) have found a necessary and sufficient condition for the zero-set of to be non-trivial with positive probability. They also provide bounds for the Hausdorff dimension of which hold with positive probability in the case that can be non-void. Here we prove that the Hausdorff dimension of is a constant almost surely on the event . Moreover, we derive a formula for the said constant. This portion of our work extends the well known formulas of Horowitz (Israel J Math 6:176–182, 1968) and Hawkes (J Lond Math Soc 8:517–525, 1974) both of which hold for one-parameter Lévy processes. More generally, we prove that for every nonrandom Borel set F in (0,∞) ^N , the Hausdorff dimension of is a constant almost surely on the event . This constant is computed explicitly in many cases. The research of N.-R. S. was supported by a grant from the Taiwan NSC.     

Projective curves of degree=codimension+2

In this article we study nondegenerate projective curves of degree d which are not arithmetically Cohen-Macaulay. Note that for a rational normal curve and a point . Our main result is about the relation between the geometric properties of X and the position of P with respect to . We show that the graded Betti numbers of X are uniquely determined by the rank of P with respect to . In particular, X satisfies property N _2,p if and only if . Therefore property N _2,p of X is controlled by and conversely can be read off from the minimal free resolution of X. This result provides a non-linearly normal example for which the converse to Theorem 1.1 in (Eisenbud et al., Compositio Math 141:1460–1478, 2005) holds. Also our result implies that for nondegenerate projective curves of degree d which are not arithmetically Cohen–Macaulay, there are exactly distinct Betti tables.     

Quasi-locally finite polynomial endomorphisms

If F is a polynomial endomorphism of , let denote the field of rational functions such that . We will say that F is quasi-locally finite if there exists a nonzero such that p(F) = 0. This terminology comes out from the fact that this definition is less restrictive than the one of locally finite endomorphisms made in Furter, Maubach (J Pure Appl Algebra 211(2):445–458, 2007). Indeed, F is called locally finite if there exists a nonzero such that p(F) = 0. In the present paper, we show that F is quasi-locally finite if and only if for each the sequence is a linear recurrent sequence. Therefore, this notion is in some sense natural. We also give a few basic results on such endomorphisms. For example: they satisfy the Jacobian conjecture.     

On <Emphasis Type="Italic">t</Emphasis><Superscript>2</Superscript>-like solutions of certain second order differential equations

Via an integral transformation, we establish two embedding results between the Emden-Fowler type equation , t ≥ t ₀ > 0, with solutions x such that as , , and the equation , u > 0, with solutions y such that for given k > 0. The conclusions of our investigation are used to derive conditions for the existence of radial solutions to the elliptic equation , , that blow up as in the two dimensional case.      

Notes on estimating inverse-Gaussian and gamma subordinators under high-frequency sampling

We study joint efficient estimation of two parameters dominating either the inverse-Gaussian or gamma subordinator, based on discrete observations sampled at satisfying as . Under the condition that as we have two kinds of optimal rates, and . Moreover, as in estimation of diffusion coefficient of a Wiener process the -consistent component of the estimator is effectively workable even when T _n does not tend to infinity. Simulation experiments are given under several h _n ’s behaviors.     

The norm of products of free random variables

Let X _i denote free identically-distributed random variables. This paper investigates how the norm of products behaves as n approaches infinity. In addition, for positive X _i it studies the asymptotic behavior of the norm of where denotes the symmetric product of two positive operators: . It is proved that if EX _i = 1, then is between and c ₂ n for certain constant c ₁ and c ₂. For it is proved that the limit of exists and equals Finally, if π is a cyclic representation of the algebra generated by X _i , and if ξ is a cyclic vector, then for all n. These results are significantly different from analogous results for commuting random variables.     

Weak convergence of diffusion processes on Wiener space

Let γ be a Gaussian measure on a Suslin space X, H be the corresponding Cameron–Martin space and {e _i } ⊂ H be an orthonormal basis of H. Suppose that μ _n = ρ _n · γ is a sequence of probability measures which converges weakly to a probability measure μ = ρ · γ Consider a sequence of Dirichlet forms , where and . We prove some sufficient conditions for Mosco convergence where . In particular, if X is a Hilbert space, and can be uniformly approximated by finite dimensional conditional expectations for every fixed e _i , then under broad assumptions Mosco and the distributions of the associated stochastic processes converge weakly.     

The symplectomorphism group of a blow up

We study the relation between the symplectomorphism group Symp M of a closed connected symplectic manifold M and the symplectomorphism and diffeomorphism groups Symp and Diff of its one point blow up . There are three main arguments. The first shows that for any oriented M the natural map from to is often injective. The second argument applies when M is simply connected and detects nontrivial elements in the homotopy group that persist into the space of self-homotopy equivalences of . Since it uses purely homological arguments, it applies to c-symplectic manifolds (M, a), that is, to manifolds of dimension 2n that support a class such that . The third argument uses the symplectic structure on M and detects nontrivial elements in the (higher) homology of BSymp, M using characteristic classes defined by parametric Gromov–Witten invariants. Some results about many point blow ups are also obtained. For example we show that if M is the four-torus with k-fold blow up (where k > 0) then is not generated by the groups as ranges over the set of all symplectic forms on . Partially supported by NSF grants DMS 0305939 and 0604769.     

A Slow Transient Diffusion in a Drifted Stable Potential

We consider a diffusion process X in a random potential of the form , where is a positive drift and is a strictly stable process of index with positive jumps. Then the diffusion is transient and converges in law towards an exponential distribution. This behaviour contrasts with the case where is a drifted Brownian motion and provides an example of a transient diffusion in a random potential which is as “slow” as in the recurrent setting.      

Dy namic of Abelia n Subgroups of GL( n, $$\mathbb{C}$$): A Structure Theorem

In this paper, we characterize the dynamic of every Abelian subgroups of , or . We show that there exists a -invariant, dense open set U in saturated by minimal orbits with a union of at most n -invariant vector subspaces of of dimension n−1 or n−2 over . As a consequence, has height at most n and in particular it admits a minimal set in . This work is supported by the research unit: systèmes dynamiques et combinatoire: 99UR15-15     

Threshold θ ≥ 2 contact processes on homogeneous trees

We study the threshold θ ≥ 2 contact process on a homogeneous tree of degree κ = b + 1, with infection parameter λ ≥ 0 and started from a product measure with density p. The corresponding mean-field model displays a discontinuous transition at a critical point and for it survives iff , where this critical density satisfies , . For large b, we show that the process on has a qualitatively similar behavior when λ is small, including the behavior at and close to the critical point . In contrast, for large λ the behavior of the process on is qualitatively distinct from that of the mean-field model in that the critical density has . We also show that , where 1 < Φ₂ < Φ₃ < ..., , and . The work of L.R.F. was partially supported by the Brazilian CNPq through grants 307978/2004-4 and 475833/2003-1, and by FAPESP through grant 04/07276-2. The work of R.H.S. was partially supported by the American N.S.F. through grant DMS-0300672.     

Normal generation and Clifford index on algebraic curves

For a smooth curve C it is known that a very ample line bundle on C is normally generated if Cliff() < Cliff(C) and there exist extremal line bundles (:non-normally generated very ample line bundle with Cliff() = Cliff(C)) with . However it has been unknown whether there exists an extremal line bundle with . In this paper, we prove that for any positive integers (g, c) with g = 2c + 5 and (mod 2) there exists a smooth curve of genus g and Clifford index c carrying an extremal line bundle with . In fact, a smooth quadric hypersurface section C of a general projective K3 surface always has an extremal line bundle with . More generally, if C has a line bundle computing the Clifford index c of C with , then C has such an extremal line bundle . For all authors, this work was supported by Korea Research Foundation Grant funded by Korea Government (MOEHRD, Basic Reasearch Promotion Fund)(KRF-2005-070-C00005).     

Rigidity and holomorphic Segre transversality for holomorphic Segre maps

Let and denote the complexifications of Heisenberg hypersurfaces in and , respectively. We show that non-degenerate holomorphic Segre mappings from into with possess a partial rigidity property. As an application, we prove that the holomorphic Segre non-transversality for a holomorphic Segre map from into with propagates along Segre varieties. We also give an example showing that this propagation property of holomorphic Segre transversality fails when N > 2n − 2.     

Kaplansky classes and derived categories

We put a monoidal model category structure on the category of chain complexes of quasi-coherent sheaves over a quasi-compact and semi-separated scheme X. The approach generalizes and simplifies the method used by the author in (Trans Am Math Soc 356(8) 3369–3390, 2004) and (Trans Am Math Soc 358(7), 2855–2874, 2006) to build monoidal model structures on the category of chain complexes of modules over a ring and chain complexes of sheaves over a ringed space. Indeed, much of the paper is dedicated to showing that in any Grothendieck category , any nice enough class of objects induces a model structure on the category Ch() of chain complexes. The main technical requirement on is the existence of a regular cardinal κ such that every object satisfies the following property: Each κ-generated subobject of F is contained in another κ-generated subobject S for which . Such a class is called a Kaplansky class. Kaplansky classes first appeared in Enochs and López-Ramos (Rend Sem Mat Univ Padova 107, 67–79, 2002) in the context of modules over a ring R. We study in detail the connection between Kaplansky classes and model categories. We also find simple conditions to put on which will guarantee that our model structure is monoidal. We will see that in several categories the class of flat objects form such Kaplansky classes, and hence induce monoidal model structures on the associated chain complex categories. We will also see that in any Grothendieck category , the class of all objects is a Kaplansky class which induces the usual (non-monoidal) injective model structure on Ch().     

Finite lattices as relative congruence lattices of finite algebras

Let A be a finite algebra and a quasivariety. By A is meant the lattice of congruences θ on A with . For any positive integer n, we give conditions on a finite algebra A under which for any n-element lattice L there is a quasivariety such that . The author was supported by INTAS grant 03-51-4110.     

A Marstrand theorem for measures with polytope density

Given and any centrally symmetric convex polytope , define we prove that if a Radon measure μ has the property then s is an integer. For the case Θ is the Euclidean ball, this result was first proved by Marstrand in 1955 for Hausdorff measure in the plane (Marstrand in Proc Lond Math Soc 3(4):257–302, 1954) and later for general Radon measures in (Marstrand in Trans Am Math Soc 205:369–392, 1964).     

Reconstruction and Subgaussian Operators in Asymptotic Geometric Analysis

We present a randomized method to approximate any vector from a set . The data one is given is the set T, vectors of and k scalar products , where are i.i.d. isotropic subgaussian random vectors in , and . We show that with high probability, any for which is close to the data vector will be a good approximation of , and that the degree of approximation is determined by a natural geometric parameter associated with the set T. We also investigate a random method to identify exactly any vector which has a relatively short support using linear subgaussian measurements as above. It turns out that our analysis, when applied to {−1, 1}-valued vectors with i.i.d. symmetric entries, yields new information on the geometry of faces of a random {−1, 1}-polytope; we show that a k- dimensional random {−1, 1}-polytope with n vertices is m-neighborly for The proofs are based on new estimates on the behavior of the empirical process when F is a subset of the L ₂ sphere. The estimates are given in terms of the γ ₂ functional with respect to the ψ ₂ metric on F, and hold both in exponential probability and in expectation. Received: November 2005, Revision: May 2006, Accepted: June 2006