Pointwise Best Approximation in the Space of Strongly Measurable Functions with Applications to Best Approximation in L(μ,X)

The object of this paper is to prove the following theorem: Let Y be a closed subspace of the Banach space X, (S,Σ,μ) a σ-finite measure space, L(S,Y) (respectively, L(S, X)) the space of all strongly measurable functions from S to Y (respectively, X), and p a positive number. Then L(S,Y) is pointwise proximinal in L(S,X) if and only if L^p(μ,Y) is proximinal in L^p(μ,X). As an application of the theorem stated above, we prove that if Y is a separable closed subspace of the Banach space X, p is a positive number, then L^p(μ,Y) is proximinal in L^p(μ,X) if and only if Y is proximinal in X. Finally, several other interesting results on pointwise best approximation are also obtained.     

From light tails to heavy tails through multiplier

Let X and Y be two independent nonnegative random variables, of which X has a distribution belonging to the class or for some γ ≥ 0 and Y is unbounded. We study how their product XY inherits the tail behavior of X. Under some mild technical assumptions we prove that the distribution of XY belongs to the class or accordingly. Hence, the multiplier Y builds a bridge between light tails and heavy tails.      

Tail behavior of the product of two dependent random variables with applications to risk theory

Let the random vector (X,Y) follow a bivariate Sarmanov distribution, where X is real-valued and Y is nonnegative. In this paper we investigate the impact of such a dependence structure between X and Y on the tail behavior of their product Z?=?XY. When X has a regularly varying tail, we establish an asymptotic formula, which extends Breiman’s theorem. Based on the obtained result, we consider a discrete-time insurance risk model with dependent insurance and financial risks, and derive the asymptotic and uniformly asymptotic behavior for the (in)finite-time ruin probabilities.     

Absolute continuity of information channels

Let [X, v, Y] be an abstract information channel with the input X = (X, ) and the output Y = (Y, ) which are measurable spaces, and denote by L(Y) = L(Y, ) the Banach space of all bounded signed measures with finite total variation as norm. The channel distribution ν(·,·) is considered as a function defined on (X, ) and valued in L(Y). It will be proved that, if the measurable space (Y, ) is countably generated, then the is a strongly measurable function from X into L(Y) if and only if there exists a probability measure μ on (Y, ) which dominates every measure ν(x, ·) (x X). Furthermore, under this condition, the Radon-Nikodym derivative ν(x, dy)/μ(dy) is jointly measurable with respect to the product measure space (X, , m) (Y, , μ) where m is any but fixed probability measure of (X, ). As an application, it will be shown that the channel given as above is uniformly approximated by channels of Hibert-Schmidt type.     

Automatic continuity of intertwining linear operators on Banach spaces

We extend the automatic continuity theory for linear operators θ:X→Y which intertwine two given bounded linear operatorsT∈L X andS∈L Y on Banach spacesX andY, respectively. This is done both by relaxing the intertwining conditionSθ=θT and by enlarging the classes of operatorsT, resp.S, well beyond the decomposable operators. Among the operatorsS captured by these extensions are multipliers on commutative semi-simple Banach algebras. dedicated to George Maltese on his 60th birthday     

Checking the adequacy of the multivariate semiparametric location shift model

Let X,X₁,…,X_m,…, Y,Y₁,…,Y_n,… be independent d-dimensional random vectors, where the X_j are i.i.d. copies of X, and the Y_k are i.i.d. copies of Y. We study a class of consistent tests for the hypothesis that Y has the same distribution as X+μ for some unspecified . The test statistic L is a weighted integral of the squared modulus of the difference of the empirical characteristic functions of and Y₁,…,Y_n, where is an estimator of μ. An alternative representation of L is given in terms of an L²-distance between two nonparametric density estimators. The finite-sample and asymptotic null distribution of L is independent of μ. Carried out as a bootstrap or permutation procedure, the test is asymptotically of a given size, irrespective of the unknown underlying distribution. A large-scale simulation study shows that the permutation procedure performs better than the bootstrap.     

Asymptotic behavior of the ratio of tail probabilities of sum and maximum of independent random variables

For a fixed integer n ≥ 2, let X ₁ ,…, X _n be independent random variables (r.v.s) with distributions F ₁,…,F _n , respectively. Let Y be another random variable with distribution G belonging to the intersection of the longtailed distribution class and the O-subexponential distribution class. When each tail of F _i , i = 1,…,n, is asymptotically less than or equal to the tail of G, we derive asymptotic lower and upper bounds for the ratio of the tail probabilities of the sum X ₁ + ⋯ + X _n and Y. By taking different G’s, we obtain general forms of some existing results.     

Equiconnected spaces and Baire classification of separately continuous functions and their analogs

We investigate the Baire classification of mappings f: X × Y → Z, where X belongs to a wide class of spaces which includes all metrizable spaces, Y is a topological space, Z is an equiconnected space, which are continuous in the first variable. We show that for a dense set in X these mappings are functions of a Baire class α in the second variable.     

Best approximation in L(μ, X), II

The object of this paper is to prove the following theorem: If Y is a closed subspace of the Banach space X, then L¹(μ, Y) is proximinal in L¹(μ, X) if and only if L^p(μ, Y) is proximinal in L^p(μ, Y) for every p, 1 < p < ∞. As an application of this result we prove that if Y is either reflexive or Y is a separable proximinal dual space, then L¹(μ, Y) is proximinal in L¹(μ, X).     

Splitting monoidal stable model categories

If C is a stable model category with a monoidal product then the set of homotopy classes of self-maps of the unit forms a commutative ring, [S,S]^C. An idempotent e of this ring will split the homotopy category: [X,Y]^C≅e[X,Y]^C⊕(1−e)[X,Y]^C. We prove that provided the localised model structures exist, this splitting of the homotopy category comes from a splitting of the model category, that is, C is Quillen equivalent to L_eSC×L_(1−e)SC and [X,Y]^L_eSC≅e[X,Y]^C. This Quillen equivalence is strong monoidal and is symmetric when the monoidal product of C is.     

Relations between ΛBV and BV(p(n) ↑∞) Classes of Functions

The base radical class L _b(X), generated by a class X was introduced in [12]. It consists of those rings whose nonzero homomorphic images have nonzero accessible subrings in X. When X is homomorphically closed, L _b(X) is the lower radical class defined by X, but otherwise X may not be contained in L _b(X). We prove that for a hereditary radical class L with semisimple class S(R), L _b(S(R)) is the class of strongly R-semisimple rings if and only if R is supernilpotent or subidempotent. A number of further examples of radical classes of the form L _b(X) are discussed. This revised version was published online in June 2006 with corrections to the Cover Date.     

Renorming in the space of Bochner integrable functionsL 1(μ,X)

A sufficient condition is given when a subspaceL⊂L ₁(μ,X) of the space of Bochner integrable function, defined on a finite and positive measure space (S, Φ, μ) with values in a Banach spaceX, is locally uniformly convex renormable in terms of the integrable evaluations {∫ _A fdμ;f∈L}. This shows the lifting property thatL ₁(μ,X) is renormable if and only ifX is, and indicates a large class of renormable subspaces even ifX does not admit and equivalent locally uniformly convex norm.     

Random difference equations with subexponential innovations

We consider the random difference equations S = _d (X + S)Y and T = _d X + TY, where = _d denotes equality in distribution, X and Y are two nonnegative random variables, and S and T on the right-hand side are independent of (X, Y). Under the assumptions that X follows a subexponential distribution with a nonzero lower Karamata index, that Y takes values in [0, 1] and is not degenerate at 0 or 1, and that (X, Y) fulfills a certain dependence structure via the conditional tail probability of X given Y, we derive some asymptotic formulas for the tail probabilities of the weak solutions S and T to these equations. In doing so we also obtain some by-products which are interesting in their own right.     

Testing for Tail Independence in Extreme Value models

Let (X,Y) be a random vector which follows in its upper tail a bivariate extreme value distribution with reverse exponentialmargins. We show that the conditional distribution function (df) of X + Y, given that X + Y>c, converges to the df F (t) = t ², , as if and only if X,Y are tail independent. Otherwise, the limit is F (t) = t. This is utilized to test for the tail independence of X, Y via various tests, including the one suggested by the Neyman–Pearson lemma. Simulations show that the Neyman–Pearson test performs best if the threshold c is close to 0, whereas otherwise it is the Kolmogorov–Smirnov test that performs best. The mathematical conditions are studied under which the Neyman–Pearson approach actually controls the type I error. Our considerations are extended to extreme value distributions in arbitrary dimensions as well as to distributions which are in a differentiable spectral neighborhood of an extreme value distribution.     

A full-invariant theorem and some applications

Let (X,τ₁) and (Y,τ₂) be two Hausdorff locally convex spaces with continuous duals X′ and Y′, respectively, L(X,Y) be the space of all continuous linear operators from X into Y, K(X,Y) be the space of all compact operators of L(X,Y). Let WOT and UOT be the weak operator topology and uniform operator topology on K(X,Y), respectively. In this paper, we characterize a full-invariant property of K(X,Y); that is, if the sequence space λ has the signed-weak gliding hump property, then each λ-multiplier WOT-convergent series ∑_iT_i in K(X,Y) must be λ-multiplier convergent with respect to all topologies between WOT and UOT if and only if each continuous linear operator T :(X,τ₁)→(λ^β,σ(λ^β,λ)) is compact. It follows from this result that the converse of Kalton's Orlicz–Pettis theorem is also true.     

A covering lemma for L(ℝ)

Jensen's celebrated Covering Lemma states that if 0^# does not exist, then for any uncountable set of ordinals X, there is a Y∈L such that X⊆Y and |X| = |Y|. Working in ZF + AD alone, we establish the following analog: If ℝ^# does not exist, then L(ℝ) and V have exactly the same sets of reals and for any set of ordinals X with |X| ≥Θ ^{L (ℝ)}, there is a Y∈L(ℝ) such that X⊆Y and |X| = |Y|. Here ℝ is the set of reals and Θ is the supremum of the ordinals which are the surjective image of ℝ. Received: 29 October 1999 / Published online: 12 December 2001     

Functions of the first Baire class with values in metrizable spaces

We show that every mapping of the first functional Lebesgue class that acts from a topological space into a separable metrizable space that is linearly connected and locally linearly connected belongs to the first Baire class. We prove that the uniform limit of functions of the first Baire class ƒ _n: X → Y belongs to the first Baire class if X is a topological space and Y is a metric space that is linearly connected and locally linearly connected. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 4, pp. 568–572, April, 2006.     

Minimal Extensions in Tensor Product Spaces

LetX,Ybe two separable Banach spaces and letVXandWYbe finite dimensional subspaces. Suppose thatVSX,WZYand letM (S, V),N (Z, W). We will prove that ifαis a reasonable, uniform crossnorm onXYthenλ_MN(V_αW,X_αY)=λ_M(V, X) λ_N(W, Y).Here for any Banach spaceX,VSXandM (S, V)

Full-size image

Also some applications of the above mentioned result will be presented.     

On isometric stability of complemented subspaces of L p

We show that the Rudin-Plotkin isometry extension theorem inL _pimplies that whenX andY are isometric subspaces ofL _pandp is not an even integer, 1≤p<∞, thenX is complemented inL _pif and only ifY is; moreover, the constants of complementation ofX andY are equal. We provide examples demonstrating that this fact fails whenp is an even integer larger than 2.     

On genera of polyhedra

We consider the stable homotopy category S of polyhedra (finite cell complexes). We say that two polyhedra X,Y are in the same genus and write X ∼ Y if X _p ≅ Y _p for all prime p, where X _p denotes the image of Xin the localized category S _p . We prove that it is equivalent to the stable isomorphism X∨B ₀ ≅Y∨B ₀, where B ₀ is the wedge of all spheres S ⁿ such that π _n ^S (X) is infinite. We also prove that a stable isomorphism X ∨ X ≅ Y ∨ X implies a stable isomorphism X ≅ Y.