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.      

On the limiting behaviour of Lévy processes at zero

We find an analytical condition characterising when the probability that a Lévy Process leaves a symmetric interval upwards goes to one as the size of the interval is shrunk to zero. We show that this is also equivalent to the probability that the process is positive at time t going to one as t goes to zero and prove some related sequential results. For each α > 0 we find an analytical condition equivalent to and as where X is a Lévy Process and T _r the time it first leaves an interval of radius r     

Limit distributions of upper order statistics for families of multivariate distributions

We consider a portfolio of dependent exchangeable random variables , where the dependence structure is generated by a mixture model (Archimedean copulas belong to this class of models). Define the ordered sample . We prove results of the following type: fix and choose appropriately, then converges in distribution to a random vector as , for which we can explicitly give the distribution.     

On the minimality of the p-harmonic ma p $$x/\|x\|$$ for weighted energy

In this paper, we study the minimality of the map for the weighted energy functional , where is a continuous function. We prove that for any integer and any non-negative, non-decreasing continuous function f, the map minimizes E _f,p among the maps in which coincide with on . The case p = 1 has been already studied in [Bourgoin J.-C. Calc. Var. (to appear)]. Then, we extend results of Hong (see Ann. Inst. Poincaré Anal. Non-linéaire 17: 35–46 (2000)). Indeed, under the same assumptions for the function f, we prove that in dimension n ≥  7 for any real with , the map minimizes E _f,p among the maps in which coincide with on .      

Laguerre geometry of hypersurfaces in $$\mathbb{R}^{n}$$

Laguerre geometry of surfaces in is given in the book of Blaschke [Vorlesungen über Differentialgeometrie, Springer, Berlin Heidelberg New York (1929)], and has been studied by Musso and Nicolodi [Trans. Am. Math. soc. 348, 4321–4337 (1996); Abh. Math. Sem. Univ. Hamburg 69, 123–138 (1999); Int. J. Math. 11(7), 911–924 (2000)], Palmer [Remarks on a variation problem in Laguerre geometry. Rendiconti di Mathematica, Serie VII, Roma, vol. 19, pp. 281–293 (1999)] and other authors. In this paper we study Laguerre differential geometry of hypersurfaces in . For any umbilical free hypersurface with non-zero principal curvatures we define a Laguerre invariant metric g on M and a Laguerre invariant self-adjoint operator : TM → TM, and show that is a complete Laguerre invariant system for hypersurfaces in with n≥ 4. We calculate the Euler–Lagrange equation for the Laguerre volume functional of Laguerre metric by using Laguerre invariants. Using the Euclidean space , the semi-Euclidean space and the degenerate space we define three Laguerre space forms , and and define the Laguerre embeddings and , analogously to what happens in the Moebius geometry where we have Moebius space forms S ⁿ , and (spaces of constant curvature) and conformal embeddings and [cf. Liu et al. in Tohoku Math. J. 53, 553–569 (2001) and Wang in Manuscr. Math. 96, 517–534 (1998)]. Using these Laguerre embeddings we can unify the Laguerre geometry of hypersurfaces in , and . As an example we show that minimal surfaces in or are Laguerre minimal in .C. Wang Partially supported by RFDP and Chuang-Xin-Qun-Ti of NSFC.     

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.     

Efficient convolution with the Newton potential in <Emphasis Type="Italic">d</Emphasis> dimensions

The paper is concerned with the evaluation of the convolution integral in d dimensions (usually d = 3), when f is given as piecewise polynomial of possibly large degree, i.e., f may be considered as an hp-finite element function. The underlying grid is locally refined using various levels of dyadically organised grids. The result of the convolution is approximated in the same kind of mesh. If f is given in tensor product form, the d-dimensional convolution can be reduced to one-dimensional convolutions. Although the details are given for the kernel the basis techniques can be generalised to homogeneous kernels, e.g., the fundamental solution of the d-dimensional Poisson equation.     

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.      

Perturbation of symmetric Markov processes

We present a path-space integral representation of the semigroup associated with the quadratic form obtained by a lower-order perturbation of the L ²-infinitesimal generator $\mathcal {L}We present a path-space integral representation of the semigroup associated with the quadratic form obtained by a lower-order perturbation of the L ²-infinitesimal generator of a general symmetric Markov process. An illuminating concrete example for is , where D is a bounded Euclidean domain in is the Laplace operator in D with zero Dirichlet boundary condition and is the fractional Laplacian in D with zero exterior condition. The strong Markov process corresponding to is a Lévy process that is the sum of Brownian motion in and an independent symmetric (2s)-stable process in killed upon exiting the domain D. This probabilistic representation is a combination of Feynman-Kac and Girsanov formulas. Crucial to the development is the use of an extension of Nakao’s stochastic integral for zero-energy additive functionals and the associated It? formula, both of which were recently developed in Chen et al. [Stochastic calculus for Dirichlet processes (preprint)(2006)]. The research of T.-S. Zhang is supported by the British EPSRC.     

The Product Formula in Unitary Deformation K-theory

For finitely generated groups G and H, we prove that there is a weak equivalence G H (G × H) of ku-algebra spectra, where denotes the “unitary deformation K-theory” functor. Additionally, we give spectral sequences for computing the homotopy groups of G and HG in terms of connective K-theory and homology of spaces of G-representations.     

Ground state alternative for <Emphasis Type="Italic">p</Emphasis>-Laplacian with potential term

Let Ω be a domain in , d ≥ 2, and 1 < p < ∞. Fix . Consider the functional Q and its Gateaux derivative Q′ given by If Q ≥ 0 on, then either there is a positive continuous function W such that for all, or there is a sequence and a function v > 0 satisfying Q′ (v) = 0, such that Q(u _k ) → 0, and in . In the latter case, v is (up to a multiplicative constant) the unique positive supersolution of the equation Q′ (u) = 0 in Ω, and one has for Q an inequality of Poincaré type: there exists a positive continuous function W such that for every satisfying there exists a constant C > 0 such that . As a consequence, we prove positivity properties for the quasilinear operator Q′ that are known to hold for general subcritical resp. critical second-order linear elliptic operators.     

Exact Distribution of the Local Score for Markovian Sequences

Let be a sequence of letters taken in a finite alphabet Θ. Let be a scoring function and the corresponding score sequence where X _i = s(A _i ). The local score is defined as follows: . We provide the exact distribution of the local score in random sequences in several models. We will first consider a Markov model on the score sequence , and then on the letter sequence . The exact P-value of the local score obtained with both models are compared thanks to several datasets. They are also compared with previous results using the independent model.     

An Extension of the Admissibility-Type Conditions for the Exponential Dichotomy of C 0-Semigroups

In the present paper we obtain a sufficient condition for the exponential dichotomy of a strongly continuous, one-parameter semigroup , in terms of the admissibility of the pair . It is already known the equivalence between the -admissibility condition and and the hyperbolicity of a C ₀-semigroup , when we assume a priori that the kernel of the dichotomic projector (denoted here by X ₂) is T(t)-invariant and is an invertible operator. We succeed to prove in this paper that the admissibility of the pair still implies the existence of an exponential dichotomy for a C ₀-semigroup even in the general case where the kernel of the dichotomic projector, X ₂, is not assumed to be T(t)-invariant.      

On closures of cycle spaces of flag domains

Open orbits D of noncompact real forms G ₀ acting on flag manifolds Z = G/Q of their semisimple complexifications G are considered. Given D and a maximal compact subgroup K ₀ of G ₀, there is a unique complex K ₀–orbit in D which is regarded as a point in the space of q-dimensional cycles in D. The group theoretical cycle space is defined to be the connected component containing C ₀ of the intersection of the G–orbit G(C ₀) with . The main result of the present article is that is closed in . This follows from an analysis of the closure of the universal domain in any G-equivariant compactification of the affine symmetric space G/K, where K is the complexification of K ₀ in G.     

A note on simultaneous Diophantine approximation on planar curves

Let denote the set of simultaneously - approximable points in and denote the set of multiplicatively ψ-approximable points in . Let be a manifold in . The aim is to develop a metric theory for the sets and analogous to the classical theory in which is simply . In this note, we mainly restrict our attention to the case that is a planar curve . A complete Hausdorff dimension theory is established for the sets and . A divergent Khintchine type result is obtained for ; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on of is full. Furthermore, in the case that is a rational quadric the convergent Khintchine type result is obtained for both types of approximation. Our results for naturally generalize the dimension and Lebesgue measure statements of Beresnevich et al. (Mem AMS, 179 (846), 1–91 (2006)). Moreover, within the multiplicative framework, our results for constitute the first of their type. The research of Victor V. Beresnevich was supported by an EPSRC Grant R90727/01. Sanju L. Velani is a Royal Society University Research Fellow. For Iona and Ayesha on No. 3.     

A local rigidity theorem for minimal surfaces in Minkowski 3-space of Randers type

Let be a Minkowski 3-space of Randers type with , where is the Euclidean metric and . We consider minimal surfaces in and prove that if a connected surface M in is minimal with respect to both the Busemann–Hausdorff volume form and the Holmes–Thompson volume form, then up to a parallel translation of , M is either a piece of plane or a piece of helicoid which is generated by lines screwing about the x ³-axis.      

Partitions of large Rado graphs

Let κ be a cardinal which is measurable after generically adding many Cohen subsets to κ and let be the κ-Rado graph. We prove, for 2 ≤ m < ω, that there is a finite value such that the set [κ] ^m can be partitioned into classes such that for any coloring of any of the classes C _i in fewer than κ colors, there is a copy of in such that is monochromatic. It follows that , that is, for any coloring of with fewer than κ colors there is a copy of such that has at most colors. On the other hand, we show that there are colorings of such that if is any copy of then for all , and hence . We characterize as the cardinality of a certain finite set of types and obtain an upper and a lower bound on its value. In particular, and for m > 2 we have where r _m is the corresponding number of types for the countable Rado graph. Research of M. Džamonja and J. A. Larson were partially supported by Engineering and Physical Sciences Research Council and research of W. J. Mitchell was partly supported by grant number DMS 0400954 from the United States National Science Foundation.     

A note on <Emphasis Type="Italic">q</Emphasis>-multiplicative functions

We prove the following statement. Let , and let . Suppose that, for all and , the sequence satisfies the relation

Stabilization of the damped wave equation with Cauchy–Ventcel boundary conditions

This paper is devoted to the study of uniform energy decay rates of solutions to the wave equation with Cauchy–Ventcel boundary conditions: