Convergence in mean of weighted sums of {an,k}-compactly uniformly integrable random elements in Banach spaces

The convergence in mean of a weighted sum ka.k(Xk EXk) of random elements in a separable Banach space is studied under a new hypothesis which relates the random elements with their respective weights in the sum: the {a.. }-compactly uniform integrability of {X. }. This condition, which is implied by the tightness of {X,,} and the {a,,k }-uniform integrability of {[IX,, II}, is weaker than the compactly miform integrability of {X,,} and leads to a result of convergence in mean which is strictly stronger than a recent result of Wang, Rao and Deli.Dirección General de Investigación Científica y TécnicaJunta de Andalucí

Lineability criteria, with applications

Lineability is a property enjoyed by some subsets within a vector space X. A subset A of X is called lineable whenever A contains, except for zero, an infinite dimensional vector subspace. If, additionally, X is endowed with richer structures, then the more stringent notions of dense-lineability, maximal dense-lineability and spaceability arise naturally. In this paper, several lineability criteria are provided and applied to specific topological vector spaces, mainly function spaces. Sometimes, such criteria furnish unified proofs of a number of scattered results in the related literature. Families of strict-order integrable functions, hypercyclic vectors, non-extendable holomorphic mappings, Riemann non-Lebesgue integrable functions, sequences not satisfying the Lebesgue dominated convergence theorem, nowhere analytic functions, bounded variation functions, entire functions with fast growth and Peano curves, among others, are analyzed from the point of view of lineability.Plan Andaluz de Investigación (Junta de Andalucía)Ministerio de Economía y Competitivida

Convergence rate of the dependent bootstrapped means

In this paper, a Baum–Katz, Erdos, Hsu–Robbins, Spitzer type complete convergence result is obtained for the dependent bootstrapped means.National Sciences and Engineering Research Council of Canad

On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry

Some techniques to show the existence and uniqueness of limit cycles, typically stated for smooth vector fields, are extended to continuous piecewise-linear differential systems. New results are obtained for systems with three linearity zones without symmetry and having one equilibrium point in the central region. We also revisit the case of systems with only two linear zones giving shorter proofs of known results.Ministerio de Ciencia e InnovaciónFondo Europeo de Desarrollo RegionalAgència de Gestió d'Ajuts Universitaris i de RecercaInstitució Catalana de Recerca i Estudis AvançatsJunta de Andalucí

On complete convergence for arrays of rowwise independent random elements

A complete convergence theorem for arrays of rowwise independent random variables was proved by Sung, Volodin, and Hu. In this paper, we extend this theorem to the Banach space without any geometric assumptions on the underlying Banach space. Our theorem also improves some known results from the literature.Korea Research Foundation Gran

Condiciones probabilísticas para la convergencia de sumas aleatoriamente ponderadas de elementos aleatorios en espacios lineales normados

"Se estudia la convergencia a cero de sumas aleatoriamente ponderadas de elementos aleatorios definidos sobre un espacio lineal normado separable sin condiciones geométricas especiales. Se investigan con especial interés las condiciones de convergencia cuando los elementos aleatorios son idénticamente distribuidos o cuando verifican una condición de acotación uniforme de momentos. Para elementos aleatorios en un espacio de Banach separable con matriz de pesos no necesariamente triangular se obtienen condiciones que establecen la equivalencia entre la convergencia en probabilidad a cero en la topología fuerte y en la topología débil. Por último se aplican en teoría de procesos estocásticos y en regresión algunos de los resultados obtenidos."

Convergence of weighted sums of random variables and uniform integrability concerning the weights

Let {ank}, n, k ∈ N be an array of real constants, and let {Xn} be a sequence of random variables. The concept of {ank}-uniform integrability of {Xn} is defined and two characterizations of this concept are established. Limit theorems for weighted sums Σk ank(Xk − EXk) are obtained, when the sequence {Xn} is {ank}-uniformly integrable

Degenerate weak convergence of row sums for arrays of random elements in stable type p Banach spaces

For an array of rowwise independent random elements {Vnj , j ≥ 1, n ≥ 1} in a real separable, stable type p Banach space X and an array of constants {anj , j ≥ 1, n ≥ 1}, general weak laws of large numbers of the forms (i) Pkn j=1 anjVnj P→ 0 and (ii) PTn j=1 anj (Vnj −cnj ) P→ 0 are obtained where for (i), EVnj = 0, j ≥ 1, n ≥ 1 and the kn are permitted to assume the value ∞ and for (ii), {cnj , j ≥ 1, n ≥ 1} is a suitable array of elements in X and {Tn, n ≥ 1} is a sequence of positive integer-valued random variables (called random indices). In the main results, the random elements {Vnj , j ≥ 1, n ≥ 1} are assumed to be stochastically dominated by a random element V and the hypotheses impose conditions on the growth behavior of the {anj , j ≥ 1, n ≥ 1}, on the tail of the distribution of ||V ||, and (for (ii)) on the marginal distributions of the random indices. The results of the form (i) are shown to be valid for a mode of convergence which is stronger than convergence in probability, viz. convergence in the Lorentz space L(p,∞)(X ). It is shown via example that the stable type p hypothesis cannot be relaxed