Strong and weak convergence theorems for asymptotically nonexpansive mappings

Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T :K→E be an asymptotically nonexpansive nonself-map with sequence {k_n}_n1[1,∞), limk_n=1, F(T):={xK: Tx=x}≠. Suppose {x_n}_n1 is generated iteratively by

where {α_n}_n1(0,1) is such that ε<1−α_n<1−ε for some ε>0. It is proved that (I−T) is demiclosed at 0. Moreover, if ∑_n1(k_n²−1)<∞ and T is completely continuous, strong convergence of {x_n} to some x^*F(T) is proved. If T is not assumed to be completely continuous but E also has a Fréchet differentiable norm, then weak convergence of {x_n} to some x^*F(T) is obtained.     

A uniform bound for the deviation of empirical distribution functions

If X₁, …, X_n are independent R^d-valued random vectors with common distribution function F, and if F_n is the empirical distribution function for X₁, …, X_n, then, among other things, it is shown that P{sup_x F_n(x) ε} 2e²(2n)^de^−2nε2 for all nε² ≥ d². The inequality remains valid if the X_i are not identically distributed and F(x) is replaced by Σ_iP{X_i ≤ x}/n.     

A Dual Approach to Constrained Interpolationfrom a Convex Subset of Hilbert Space

Many interesting and important problems of best approximationare included in (or can be reduced to) one of the followingtype: in a Hilbert spaceX, find the best approximationP_K(x) to anyxXfrom the setKC∩A⁻¹(b),whereCis a closed convex subset ofX,Ais a bounded linearoperator fromXinto a finite-dimensional Hilbert spaceY, andbY. The main point of this paper is to show thatP_K(x)isidenticaltoP_C(x+A*y)—the best approximationto a certain perturbationx+A*yofx—from the convexsetCor from a certain convex extremal subsetC_bofC. Thelatter best approximation is generally much easier to computethan the former. Prior to this, the result had been known onlyin the case of a convex cone or forspecialdata sets associatedwith a closed convex set. In fact, we give anintrinsic characterizationof those pairs of setsCandA⁻¹(b) for which this canalways be done. Finally, in many cases, the best approximationP_C(x+A*y) can be obtained numerically from existingalgorithms or from modifications to existing algorithms. Wegive such an algorithm and prove its convergence     

On the approximate behavior of the posterior distribution for an extreme multivariate observation

The behavior of the posterior for a large observation is considered. Two basic situations are discussed; location vectors and natural parameters.Let X = (X₁, X₂, …, X_n) be an observation from a multivariate exponential distribution with that natural parameter Θ = (Θ₁, Θ₂, …, Θ_n). Let θ_x^* be the posterior mode. Sufficient conditions are presented for the distribution of Θ − θ_x^* given X = x to converge to a multivariate normal with mean vector 0 as |x| tends to infinity. These same conditions imply that E(Θ | X = x) − θ_x^* converges to the zero vector as |x| tends to infinity.The posterior for an observation X = (X₁, X₂, …, X_n is considered for a location vector Θ = (Θ₁, Θ₂, …, Θ_n) as x gets large along a path, γ, in Rⁿ. Sufficient conditions are given for the distribution of γ(t) − Θ given X = γ(t) to converge in law as t → ∞. Slightly stronger conditions ensure that γ(t) − E(Θ | X = γ(t)) converges to the mean of the limiting distribution.These basic results about the posterior mean are extended to cover other estimators. Loss functions which are convex functions of absolute error are considered. Let δ be a Bayes estimator for a loss function of this type. Generally, if the distribution of Θ − E(Θ | X = γ(t)) given X = γ(t) converges in law to a symmetric distribution as t → ∞, it is shown that δ(γ(t)) − E(Θ | X = γ(t)) → 0 as t → ∞.     

The Central Limit Theorem for Free Additive Convolution

Let {X_n}^∞_n=1be a sequence of free, identically distributed random variables with common distributionμ. Then there exist sequences {B_n}^∞_n=1and {A_n}^∞_n=1of positive and real numbers, respectively, such that sequence of random variables[formula]converges in distribution to the semicircle law if and only if the function[formula]is slowly varying in Karamata's sense. In other words, the free domain of attraction of the semicircle law coincides with the classical domain of attraction of the Gaussian. We prove an analogous result for normal domains of attraction in the sense of Linnik.     

Set-Indexed Conditional Empirical and Quantile Processes Based on Dependent Data

We consider a conditional empirical distribution of the form F_n(C x)=∑ⁿ_t=1 ω_n(X_t−x) I{Y_tC} indexed by C , where {(X_t, Y_t), t=1, …, n} are observations from a strictly stationary and strong mixing stochastic process, {ω_n(X_t−x)} are kernel weights, and is a class of sets. Under the assumption on the richness of the index class in terms of metric entropy with bracketing, we have established uniform convergence and asymptotic normality for F_n(· x). The key result specifies rates of convergences for the modulus of continuity of the conditional empirical process. The results are then applied to derive Bahadur–Kiefer type approximations for a generalized conditional quantile process which, in the case with independent observations, generalizes and improves earlier results. Potential applications in the areas of estimating level sets and testing for unimodality (or multimodality) of conditional distributions are discussed.     

Implicit vector equilibrium problems with applications

Let X and Y be Hausdorff topological vector spaces, K a nonempty, closed, and convex subset of X, C : K → 2^Y a point-to-set mapping such that for any χ ε K, C(χ) is a pointed, closed, and convex cone in Y and int C(χ) ≠ 0. Given a mapping g : K → K and a vector valued bifunction f : K × K → Y, we consider the implicit vector equilibrium problem (IVEP) of finding χ^* ε K such that f g(χ^*), y) -int C(χ) for all y ε K. This problem generalizes the (scalar) implicit equilibrium problem and implicit variational inequality problem. We propose the dual of the implicit vector equilibrium problem (DIVEP) and establish the equivalence between (IVEP) and (DIVEP) under certain assumptions. Also, we give characterizations of the set of solutions for (IVP) in case of nonmonotonicity, weak C-pseudomonotonicity, C-pseudomonotonicity, and strict C-pseudomonotonicity, respectively. Under these assumptions, we conclude that the sets of solutions are nonempty, closed, and convex. Finally, we give some applications of (IVEP) to vector variational inequality problems and vector optimization problems.     

On almost Chebyshev subspaces

A closed subspace F in a Banach space X is called almost Chebyshev if the set of x ε X which fail to have unique best approximation in F is contained in a first category subset. We prove, among other results, that if X is a separable Banach space which is either locally uniformly convex or has the Radon-Nikodym property, then “almost all” closed subspaces are almost Chebyshev.     

Generalized Hyers–Ulam–Rassias stability of n-sesquilinear-quadratic mappings on Banach modules over -algebras

Assume that X is a left Banach module over a unital C^*-algebra A. It is shown that almost every n-sesquilinear-quadratic mapping h:X×X×Xⁿ→A is an n-sesquilinear-quadratic mapping when holds for all x,y,z₁,…,z_nX.Moreover, we prove the generalized Hyers–Ulam–Rassias stability of an n-sesquilinear-quadratic mapping on a left Banach module over a unital C^*-algebra.     

Multifunctions of faces for conditional expectations of selectors and Jensen's inequality

Let (T, , P) be a probability space, a P-complete sub-δ-algebra of and X a Banach space. Let multifunction t → Γ(t), t T, have a (X)-measurable graph and closed convex subsets of X for values. If x(t) ε Γ(t) P-a.e. and y(·) ε E_p x(·), then y(t) ε Γ(t) P-a.e. Conversely, x(t) ε F(Γ(t), y(t)) P-a.e., where F(Γ(t), y(t)) is the face of point y(t) in Γ(t). If X = , then the same holds true if Γ(t) is Borel and convex, only. These results imply, in particular, extensions of Jensen's inequality for conditional expectations of random convex functions and provide a complete characterization of the cases when the equality holds in the extended Jensen inequality.     

Algébres topologiquement uniformes et algébres fortement topologiquement uniformes

We characterize locally convex topological algebrasA satisfying: a sequence (x _n) inA converges to 0 if, and only if, (x _n ²) converges to 0. We also show that a real Banach algebra such thatx _n ²+y _n ²→0 if, and only if,x _n → 0 andy _n → 0, for every sequences (x _n) and (y _n) inA, is isomorphic to, whereX is a compact space.      

Special Uniform Approximations of Continuous Vector-Valued Functions. Part I: Special Approximations in CX(T)

In this paper, we give special uniform approximations of functions u from the spaces C_X(T) and C_∞(T,X), with elements of the tensor products C_Γ(T)X, respectively C₀(T,Γ)X, for a topological space T and a Γ-locally convex space X. We call an approximation special, if satisfies additional constraints, namely supp vu⁻¹(X\{0}) and (T) co(u(T)) (resp. co(u(T){0})). In Section 3, we give three distinct applications, which are due exactly to these constraints: a density result with respect to the inductive limit topology, a Tietze–Dugundji's type extension new theorem and a proof of Schauder–Tihonov's fixed point theorem.     

Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces

This paper studies the convergence of the sequence defined by x0∈C,xn 1=αnu (1-αn)Txn,n=0,1,2,…, where 0 ≤αn ≤ 1, limn→∞αn = 0, ∑∞n=0 αn = ∞, and T is a nonexpansive mapping from a nonempty closed convex subset C of a Banach space X into itself. The iterative sequence {xn} converges strongly to a fixed point of T in the case when X is a uniformly convex Banach space with a uniformly Gateaux differentiable norm or a uniformly smooth Banach space only. The results presented in this paper extend and improve some recent results.     

Minimax estimation of the mean of spherically symmetric distributions under general quadratic loss

For X one observation on a p-dimensional (p ≥ 4) spherically symmetric (s.s.) distribution about θ, minimax estimators whose risks dominate the risk of X (the best invariant procedure) are found with respect to general quadratic loss, L(δ, θ) = (δ − θ)′ D(δ − θ) where D is a known p × p positive definite matrix. For C a p × p known positive definite matrix, conditions are given under which estimators of the form δ_a,r,C,D(X) = (I − (ar(|X|²)) D^−1/2CD^1/2 |X|⁻²)X are minimax with smaller risk than X. For the problem of estimating the mean when n observations X₁, X₂, …, X_n are taken on a p-dimensional s.s. distribution about θ, any spherically symmetric translation invariant estimator, δ(X₁, X₂, …, X_n), with have a s.s. distribution about θ. Among the estimators which have these properties are best invariant estimators, sample means and maximum likelihood estimators. Moreover, under certain conditions, improved robust estimators can be found.     

On Jakovlev spaces

A topological spaceX is called weakly first countable, if for every pointx there is a countable family {C _n ^x |n ∈ω} such thatx ∈C _n ^+1x ⊆C _n ^x and such thatU ⊂X is open iff for eachx ∈U someC _n ^x is contained inU. This weakening of first countability is due to A. V. Arhangelskii from 1966, who asked whether compact weakly first countable spaces are first countable. In 1976, N. N. Jakovlev gave a negative answer under the assumption of continuum hypothesis. His result was strengthened by V. I. Malykhin in 1982, again under CH. In the present paper we construct various Jakovlev type spaces under the weaker assumption b=c, and also by forcing. The second author was supported by the Ben-Gurion University Center for Advanced Studies in Mathematics, Be’er Sheva. The third author was supported by OTKA grant no. 37758 of Hungary.     

Centres of Convex Sets inLMetrics

It is shown that for each convex bodyARⁿthere exists a naturally defined family _AC(Sⁿ⁻¹) such that for everyg _A, and every convex functionf: R→Rthe mappingy∫_Sⁿ⁻¹ f(g(x)−y, x) dσ(x) has a minimizer which belongs toA. As an application, approximation of convex bodies by balls with respect toL^pmetrics is discussed.     

A logarithmic Gauss curvature flow and the Minkowski problem

Let X₀ be a smooth uniformly convex hypersurface and f a postive smooth function in Sⁿ. We study the motion of convex hypersurfaces X(·,t) with initial X(·,0)=θX₀ along its inner normal at a rate equal to log(K/f) where K is the Gauss curvature of X(·,t). We show that the hypersurfaces remain smooth and uniformly convex, and there exists θ^*>0 such that if θ<θ^*, they shrink to a point in finite time and, if θ>θ^*, they expand to an asymptotic sphere. Finally, when θ=θ^*, they converge to a convex hypersurface of which Gauss curvature is given explicitly by a function depending on f(x).     

A Chebyshev Set and its Distance Function

We prove that in a Banach space X with rotund dual X^* a Chebyshev set C is convex iff the distance function d_C is regular on X\C iff d_C admits the strict and Gâteaux derivatives on X\C which are determined by the subdifferential ∂x−

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x" height="11" width="10"> for each xX\C and

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x" height="11" width="10">P_C(x)\{cC:x−c=d_C(x)}. If X is a reflexive Banach space with smooth and Kadec norm then C is convex iff it is weakly closed iff P_C is continuous. If the norms of X and X^* are Fréchet differentiable then C is convex iff d_C is Fréchet differentiable on X\C. If also X has a uniformly Gâteaux differentiable norm then C is convex iff the Gâteaux (Fréchet) subdifferential ∂⁻d_C(x) (∂_Fd_C(x)) is nonempty on X\C.     

An Asymptotic Property of Schachermayer's Space under Renorming

Let X be a Banach space with closed unit ball B. Given k , X is said to be k-β, respectively, (k + 1)-nearly uniformly convex ((k + 1)-NUC), if for every ε > 0 there exists δ, 0 < δ < 1, so that for every x B and every ε-separated sequence (x_n) B there are indices (n_i)^k_{i = 1}, respectively, (n_i)^{k + 1}_{i = 1}, such that (1/(k + 1))||x + ∑^k_{i = 1} x_ni|| ≤ 1 − δ, respectively, (1/(k + 1))||∑^{k + 1}_{i = 1} x_ni|| ≤ 1 − δ. It is shown that a Banach space constructed by Schachermayer is 2-β, but is not isomorphic to any 2-NUC Banach space. Modifying this example, we also show that there is a 2-NUC Banach space which cannot be equivalently renormed to be 1-β.     

Limiting -subgradient characterizations of constrained best approximation

In this paper, we show that the strong conical hull intersection property (CHIP) completely characterizes the best approximation to any x in a Hilbert space X from the set

K:=C∩{xX:-g(x)S},