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.      

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.     

Density of extremal measures in parabolic potential theory

It is shown that, for the heat equation on , d ≥ 1, any convex combination of harmonic (= caloric) measures , where U ₁, . . . , U _k are relatively compact open neighborhoods of a given point x, can be approximated by a sequence of harmonic measures such that each W _n is an open neighborhood of x in . Moreover, it is proven that, for every open set U in containing x, the extremal representing measures for x with respect to the convex cone of potentials on U (these measures are obtained by balayage, with respect to U, of the Dirac measure at x on Borel subsets of U) are dense in the compact convex set of all representing measures. Since essential ingredients for a proof of corresponding results in the classical case (or more general elliptic situations; see Hansen and Netuka in Adv. Math. 218(4):1181–1223, 2008) are not available for the heat equation, an approach heavily relying on the transit character of the hyperplanes , , is developed. In fact, the new method is suitable to obtain convexity results for limits of harmonic measures and the density of extremal representing measures on for practically every space–time structure which is given by a sub-Markov semigroup (P _t ) _t>0 on a space X′ such that there are strictly positive continuous densities with respect to a (non-atomic) measure on X′. In particular, this includes many diffusions and corresponding symmetric processes given by heat kernels on manifolds and fractals. Moreover, the results may be applied to restrictions of the space–time structure on arbitrary open subsets. I. Netuka’s research was supported in part by the project MSM 0021620839 financed by MSMT, by the grant 201/07/0388 of the Grant Agency of the Czech Republic, and by CRC-701, Bielefeld.     

Bubble accumulations in an elliptic Neumann problem with critical Sobolev exponent

We consider the following critical elliptic Neumann problem on , Ω; being a smooth bounded domain in is a large number. We show that at a positive nondegenerate local minimum point Q ₀ of the mean curvature (we may assume that Q ₀ = 0 and the unit normal at Q ₀ is − e _N ) for any fixed integer K ≥ 2, there exists a μ _K > 0 such that for μ > μ _K , the above problem has K−bubble solution u _μ concentrating at the same point Q ₀. More precisely, we show that u _μ has K local maximum points Q ₁^μ, ... , Q _K ^μ ∈∂Ω with the property that and approach an optimal configuration of the following functional (*) Find out the optimal configuration that minimizes the following functional: where are two generic constants and φ (Q) = Q ^T G Q with G = (∇ _ij H(Q ₀)). Research supported in part by an Earmarked Grant from RGC of HK.     

On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems

We consider the computation of stable approximations to the exact solution of nonlinear ill-posed inverse problems F(x) = y with nonlinear operators F : X → Y between two Hilbert spaces X and Y by the Newton type methods

Continuous spectrum for a class of nonhomogeneous differential operators

We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in with smooth boundary, λ is a positive real number, and the continuous functions p ₁, p ₂, and q satisfy 1 < p ₂(x) < q(x) < p ₁(x) < N and for any . The main result of this paper establishes the existence of two positive constants λ₀ and λ₁ with λ₀ ≤ λ₁ such that any is an eigenvalue, while any is not an eigenvalue of the above problem.     

Cyclicity of bicyclic operators and completeness of translates

We study cyclicity of operators on a separable Banach space which admit a bicyclic vector such that the norms of its images under the iterates of the operator satisfy certain growth conditions. A simple consequence of our main result is that a bicyclic unitary operator on a Banach space with separable dual is cyclic. Our results also imply that if is the shift operator acting on the weighted space of sequences , if the weight ω satisfies some regularity conditions and ω(n) = 1 for nonnegative n, then S is cyclic if . On the other hand one can see that S is not cyclic if the series diverges. We show that the question of Herrero whether either S or S* is cyclic on admits a positive answer when the series is convergent. We also prove completeness results for translates in certain Banach spaces of functions on .     

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     

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.     

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.      

Analysis of degenerate elliptic operators of Grušin type

We analyse degenerate, second-order, elliptic operators H in divergence form on L ₂(R ⁿ  × R ^m ). We assume the coefficients are real symmetric and a ₁ H _δ  ≥ H ≥ a ₂ H _δ for some a ₁, a ₂ > 0 where

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.     

On a Poincaré-type Inequality for Energy Forms in L p

We consider Dirichlet spaces ( ) in L ² and more general energy forms in L ^p , . For the latter we introduce the notions of an extended ’Dirichlet’ space and a transient form. Under the assumption that , resp. , are compactly embedded in L ², resp. L ^p , we prove a Poincaré inequality for transient (Dirichlet) forms. If both and its adjoint are sub-Markovian semigroups, we show that the transience of T _t is independent of ) and that it is implied by the transience of the energy form of and the form belonging to .     

Solutions of elliptic problems with nonlinearities of linear growth

In this paper, we study existence of nontrivial solutions to the elliptic equation

A short proof for the sum formula and its generalization

For positive integers with a _r  = 2, the multiple zeta value or r-fold Euler sum is defined as [2]

Random walk in random scenery and self-intersection local times in dimensions d ≥ 5

Let {S _k , k ≥ 0} be a symmetric random walk on , and an independent random field of centered i.i.d. random variables with tail decay . We consider a random walk in random scenery, that is . We present asymptotics for the probability, over both randomness, that {X _n > n ^β} for β > 1/2 and α > 1. To obtain such asymptotics, we establish large deviations estimates for the self-intersection local times process , where l _n (x) is the number of visits of site x up to time n.      

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.     

Boundary Harnack principle for Brownian motions with measure-valued drifts in bounded Lipschitz domains

Let be such that each is a signed measure on R ^d belonging to the Kato class K _{d, 1}. A Brownian motion in R ^d with drift is a diffusion process in R ^d whose generator can be informally written as . When each is given by U ⁱ (x)dx for some function U ⁱ , a Brownian motion with drift is a diffusion in R ^d with generator . In Kim and Song (Ill J Math 50(3):635–688, 2006), some properties of Brownian motions with measure-value drifts in bounded smooth domains were discussed. In this paper we prove a scale invariant boundary Harnack principle for the positive harmonic functions of Brownian motions with measure-value drifts in bounded Lipschitz domains. We also show that the Martin boundary and the minimal Martin boundary with respect to Brownian motions with measure-valued drifts coincide with the Euclidean boundary for bounded Lipschitz domains. The results of this paper are also true for diffusions with measure-valued drifts, that is, when is replaced by a uniformly elliptic divergence form operator with C ¹ coefficients or a uniformly elliptic non-divergence form operator with C ¹ coefficients. The research of R. Song is supported in part by a joint US-Croatia grant INT 0302167. The research of P. Kim is supported by Research Settlement Fund for the new faculty of Seoul National University.     

Affine restriction for radial surfaces

Suppose dμ is affine surface measure on a convex radial surface Γ(x) = (x, γ(|x|)), a ≤ |x| < b, in . Under appropriate smoothness and growth conditions on γ, we prove and Fourier restriction estimates for Γ.     

Compactness results for divergence type nonlinear elliptic equations

Let M be a compact manifold of dimension n ≥ 2 and 1 < p < n. For a family of functions F _α defined on TM, which are p-homogeneous, positive, and convex on each fiber, of Riemannian metrics g _α and of coefficients a _α on M, we discuss the compactness problem of minimal energy type solutions of the equation