Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userFrom convergence in distribution to uniform convergence
73
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We present conditions that allow us to pass from the convergence of probability measures in distribution to the uniform convergence of the associated quantile functions. Under these conditions, one can in particular pass from the asymptotic distribution of collections of real numbers, such as the eigenvalues of a family of $n$-by-$n$ matrices as $n$ goes to infinity, to their uniform approximation by the values of the quantile function at equidistant points. For Hermitian Toeplitz-like matrices, convergence in distribution is ensured by theorems of the SzegH{o} type. Our results transfer these convergence theorems into uniform convergence statements.
rate research
Read More
Necessary and sufficient conditions are presented for the Abel averages of discrete and strongly continuous semigroups, $T^k$ and $T_t$, to be power convergent in the operator norm in a complex Banach space. These results cover also the case where $T$ is unbounded and the corresponding Abel average is defined by means of the resolvent of $T$. They complement the classical results by Michael Lin establishing sufficient conditions for the corresponding convergence for a bounded $T$.
In this paper we look for the existence of large linear and algebraic structures of sequences of measurable functions with different modes of convergence. Concretely, the algebraic size of the family of sequences that are convergent in measure but not a.e.~pointwise, uniformly but not pointwise convergent, and uniformly convergent but not in $L^1$-norm, are analyzed. These findings extend and complement a number of earlier results by several authors.
In the present paper the unconditional convergence and the invertibility of multipliers is investigated. Multipliers are operators created by (frame-like) analysis, multiplication by a fixed symbol, and resynthesis. Sufficient and/or necessary conditions for unconditional convergence and invertibility are determined depending on the properties of the analysis and synthesis sequences, as well as the symbol. Examples which show that the given assertions cover different classes of multipliers are given. If a multiplier is invertible, a formula for the inverse operator is determined. The case when one of the sequences is a Riesz basis is completely characterized.
Given an ideal $mathcal{I}$ on the positive integers, a real sequence $(x_n)$ is said to be $mathcal{I}$-statistically convergent to $ell$ provided that $$ textstyle left{n in mathbf{N}: frac{1}{n}|{k le n: x_k otin U}| ge varepsilonright} in mathcal{I} $$ for all neighborhoods $U$ of $ell$ and all $varepsilon>0$. First, we show that $mathcal{I}$-statistical convergence coincides with $mathcal{J}$-convergence, for some unique ideal $mathcal{J}=mathcal{J}(mathcal{I})$. In addition, $mathcal{J}$ is Borel [analytic, coanalytic, respectively] whenever $mathcal{I}$ is Borel [analytic, coanalytic, resp.]. Then we prove, among others, that if $mathcal{I}$ is the summable ideal ${Asubseteq mathbf{N}: sum_{a in A}1/a<infty}$ or the density zero ideal ${Asubseteq mathbf{N}: lim_{nto infty} frac{1}{n}|Acap [1,n]|=0}$ then $mathcal{I}$-statistical convergence coincides with statistical convergence. This can be seen as a Tauberian theorem which extends a classical theorem of Fridy. Lastly, we show that this is never the case if $mathcal{I}$ is maximal.
Unbounded order convergence has lately been systematically studied as a generalization of almost everywhere convergence to the abstract setting of vector and Banach lattices. This paper presents a duality theory for unbounded order convergence. We define the unbounded order dual (or uo-dual) $X_{uo}^sim$ of a Banach lattice $X$ and identify it as the order continuous part of the order continuous dual $X_n^sim$. The result allows us to characterize the Banach lattices that have order continuous preduals and to show that an order continuous predual is unique when it exists. Applications to the Fenchel-Moreau duality theory of convex functionals are given. The applications are of interest in the theory of risk measures in Mathematical Finance.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
