On Convergence and Convolutions of Random Signed Measures

Let μ _n be a sequence of random finite signed measures on the locally compact group G equal to either or ℝ ^d . We give weak conditions on the sequence μ _n and on functions K such that the convolution product μ _n *K, and its derivatives, converge in law, in probability, or almost surely in the Banach spaces or L ^p (G). Examples for sequences μ _n covered are the empirical process (possibly arising from dependent data) and also random signed measures where is some (nonparametric) estimator for the measure ℙ, including the usual kernel and wavelet based density estimators with MISE-optimal bandwidths. As a statistical application, we apply the results to study convolutions of density estimators.      

( m,n)-Equidista nt Sets i n $\ mathbb{R}^{k},\ mathbb{S}^{k}$ , a nd ℙk

We call a metric space X (m,n)-equidistant if, when A⊆X has exactly m points, there are exactly n points in X each of which is equidistant from (the points of) A. We prove that, for k≥2, the Euclidean space ℝ ^k contains an (m,1)-equidistant set if and only if k≥m. Although the sphere is (3,2)-equidistant, and ℝ⁴ contain no (4,2)-equidistant sets. We discuss related results about projective spaces, and state a conjecture about analogous to the Double Midset Conjecture.     

A Characterization of Quadric Constant Mean Curvature Hypersurfaces of Spheres

Let be an immersion of a complete n-dimensional oriented manifold. For any v∈ℝ ⁿ⁺², let us denote by ℓ _v :M→ℝ the function given by ℓ _v (x)=〈φ(x),v〉 and by f _v :M→ℝ, the function given by f _v (x)=〈ν(x),v〉, where is a Gauss map. We will prove that if M has constant mean curvature, and, for some v≠0 and some real number λ, we have that ℓ _v =λ f _v , then, φ(M) is either a totally umbilical sphere or a Clifford hypersurface. As an application, we will use this result to prove that the weak stability index of any compact constant mean curvature hypersurface M ⁿ in which is neither totally umbilical nor a Clifford hypersurface and has constant scalar curvature is greater than or equal to 2n+4. A. Brasil Jr. was partially supported by CNPq, Brazil, 306626/2007-1.     

On normal approximation of discounted and strongly mixing random variables

We estimate the difference for bounded functions h: ℝ → ℝ satisfying the Lipschitz condition, where Z _v = B _v ⁻¹ ∑ _i=0 ^∞ v ⁱ X _i and with discount factor ν such that 0 < ν < 1. Here {X _n , n ≥ 0} is a sequence of strongly mixing random variables with , and N is a standard normal random variable. In a particular case, the obtained upper bounds are of order O((1 − ν)^1/2). Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 3, pp. 399–409, July–September, 2007. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-15/07.     

Bounded universal functions for sequences of holomorphic self-maps of the disk

We give several characterizations of those sequences of holomorphic self-maps {φ _n } _n≥1 of the unit disk for which there exists a function F in the unit ball of H ^∞ such that the orbit {F∘φ _n :n∈ℕ} is locally uniformly dense in . Such a function F is said to be a -universal function. One of our conditions is stated in terms of the hyperbolic derivatives of the functions φ _n . As a consequence we will see that if φ _n is the nth iterate of a map φ of into , then {φ _n } _n≥1 admits a -universal function if and only if φ is a parabolic or hyperbolic automorphism of . We show that whenever there exists a -universal function, then this function can be chosen to be a Blaschke product. Further, if there is a -universal function, we show that there exist uniformly closed subspaces consisting entirely of universal functions.     

Multilinear Singular Integrals with Rough Kernel

For a class of multilinear singular integral operators T _A ,

Sharp estimates for maximal operators associated to the wave equation

The wave equation, ∂ _tt u=Δu, in ℝ ⁿ⁺¹, considered with initial data u(x,0)=f∈H ^s (ℝ ⁿ ) and u’(x,0)=0, has a solution which we denote by . We give almost sharp conditions under which and are bounded from H ^s (ℝ ⁿ ) to L ^q (ℝ ⁿ ).     

Relations between different types of spectra and spectral characterizations

In the study of the asymptotic behaviour of solutions of differential-difference equations the -spectrum has been useful, where and implies Fourier transform , with given , φ∈L ^∞(ℝ,X), X a Banach space, (half)line. Here we study and related concepts, give relations between them, especially weak Laplace half-line spectrum of φ, and thus ⊂ classical Beurling spectrum = Carleman spectrum =  ; also  = Beurling spectrum of “φ modulo ” (Chill-Fasangova). If satisfies a Loomis type condition (L _U ), then countable and uniformly continuous ∈U are shown to imply ; here (L _U ) usually means , indefinite integral Pf of f in U imply Pf in (the Bohl-Bohr theorem for = almost periodic functions, U=bounded functions). This spectral characterization and other results are extended to unbounded functions via mean classes , ℳ ^m U ((2.1) below) and even to distributions, generalizing various recent results for uniformly continuous bounded φ. Furthermore for solutions of convolution systems S*φ=b with in some we show . With these above results, one gets generalizations of earlier results on the asymptotic behaviour of solutions of neutral integro-differential-difference systems. Also many examples and special cases are discussed.     

Diophantine exponents for mildly restricted approximation

We are studying the Diophantine exponent μ _n,l defined for integers 1≤l<n and a vector α∈ℝ ⁿ by letting

Geometry of finite-dimensional normed spaces,and continuous functions on the Euclidean sphere

Let ℝⁿ be the n-dimensional Euclidean space, and let { · } be a norm in Rⁿ. Two lines ℓ₁ and ℓ₂ in ℝⁿ are said to be { · }-orthogonal if their { · }-unit direction vectors e ₁ and e ₂ satisfy {e ₁ + e ₂} = {e ₁ − e ₂}. It is proved that for any two norms { · } and { · }′ in ℝⁿ there are n lines ℓ₁, ..., ℓ_n that are { · }-and { · }′-orthogonal simultaneously. Let be a continuous function on the unit sphere with center O. It is proved that there exists an (n − 1)-cube C centered at O, inscribed in , and such that all sums of values of f at the vertices of (n − 3)-faces of C are pairwise equal. If the function f is even, then there exists an n-cube with the same properties. Furthermore, there exists an orthonormal basis e ₁, ..., e _n such that for 1 ≤ i ≤ j ≤ n we have . Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 107–117.     

Limit Theorems for Sums of Heavy-tailed Variables with Random Dependent Weights

Let be independent and identically distributed random variables with heavy-tailed distributions. Consider a sequence of random weights , independent of and focus on the weighted sums , where μ involves a suitable centering. We establish sufficient conditions for these weighted sums to converge to non-trivial limit processes, as n→∞, when appropriately normalized. The convergence holds, for example, if is strictly stationary, dependent, and W ₁ has lighter tails than U ₁. In particular, the weights W _j s can be strongly dependent. The limit processes are scale mixtures of stable Lévy motions. We establish weak convergence in the Skorohod J ₁-topology. We also consider multivariate weights and show that they converge weakly in the strong Skorohod M ₁-topology. The M ₁-topology, while weaker than the J ₁-topology, is strong enough for the supremum and infimum functionals to be continuous. This research was partially supported by a fellowship of the Horace H. Rackham School of Graduate Studies at the University of Michigan and the NSF Grants BCS-0318209 and DMS-0505747 at Boston University.     

Boundedness of Some Marcinkiewicz Integral Operators Related to Higher Order Commutators on Hardy Spaces

In this paper, the authors study the boundedness properties of μΩ↑m,b generated by the function b ∈Lipβ（R^n）（0 〈β≤ 1/m） and the Marcinkiewicz integrals operator μΩ. The boundednesses are established on the Hardy type spaces Hb^m^p,n（R^n） and the Herz Hardy type spaces Hbm Kq^α,p（R^b）.     

A central limit theorem for convex sets

We show that there exists a sequence for which the following holds: Let K⊂ℝⁿ be a compact, convex set with a non-empty interior. Let X be a random vector that is distributed uniformly in K. Then there exist a unit vector θ in ℝⁿ, t₀∈ℝ and σ>0 such that

Line Transversals to Disjoint Balls

We prove that the set of directions of lines intersecting three disjoint balls in ℝ³ in a given order is a strictly convex subset of . We then generalize this result to n disjoint balls in ℝ ^d . As a consequence, we can improve upon several old and new results on line transversals to disjoint balls in arbitrary dimension, such as bounds on the number of connected components and Helly-type theorems.     

Approximation of bandlimited functions by finite exponential sums

For a compact set K in ℝ ⁿ , let B ² _K be the set of all functions f ∈ L ²(ℝ²) bandlimited to K, i.e., such that the Fourier transform f̂ of f is supported by K. We investigate the question of approximation of f ∈ B ² _K by finite exponential sums

Dilation of Newton Polytope and <Emphasis Type="Italic">p</Emphasis>-adic Estimate

Let f(X) be a polynomial in n variables over the finite field  \mathbbF_q\mathbb{F}_{q}. Its Newton polytope Δ(f) is the convex closure in ℝ ⁿ of the origin and the exponent vectors (viewed as points in ℝ ⁿ ) of monomials in f(X). The minimal dilation of Δ(f) such that it contains at least one lattice point of $\mathbb{Z}_{>0}^{n}$\mathbb{Z}_{>0}^{n} plays a vital pole in the p-adic estimate of the number of zeros of f(X) in  \mathbbF_q\mathbb{F}_{q}. Using this fact, we obtain several tight and computational bounds for the dilation which unify and improve a number of previous results in this direction.     

Plurisubharmonic Functions on the Octonionic Plane and <Emphasis Type="Italic">Spin</Emphasis>(9)-Invariant Valuations on Convex Sets

A new class of plurisubharmonic functions on the octonionic plane is introduced. An octonionic version of theorems of A.D. Aleksandrov (Vestnik Leningrad. Univ. Ser. Mat. Meh. Astr. 13(1):5–24, 1958) and Chern-Levine-Nirenberg (Global Analysis, pp. 119–139, 1969), and Błocki (Proc. Am. Math. Soc. 128(12):3595–3599, 2000) are proved. These results are used to construct new examples of continuous translation invariant valuations on convex subsets of . In particular, a new example of Spin(9)-invariant valuation on ℝ¹⁶ is given. Partially supported by ISF grant 1369/04.     

Quasiconvexification of geometric integrals

We study the existence of an integral representation for the functional

On the Number of Topological Types Occurring in a Parameterized Family of Arrangements

Let be an o-minimal structure over ℝ, a closed definable set, and