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Tightness and π-character in centered spaces

100%

Bell M.

Colloquium Mathematicum

1999

tom 80

nr 2

297-307

We continue an investigation into centered spaces, a generalization of dyadic spaces. The presence of large Cantor cubes in centered spaces is deduced from tightness considerations. It follows that for centered spaces X, πχ(X) = t(X), and if X has uncountable tightness, then t(X) = sup{κ : $2^κ$ ⊂ X}. The relationships between 9 popular cardinal functions for the class of centered spaces are justified. An example is constructed which shows, unlike the dyadic and polyadic properties, that the centered property is not preserved by passage to a zeroset.

Weak Hölder convergence of processes with application to the perturbed empirical process

75%

Hamadouche D., Suquet C.

Applicationes Mathematicae

1999

tom 26

nr 1

63-83

We consider stochastic processes as random elements in some spaces of Hölder functions vanishing at infinity. The corresponding scale of spaces $C^{α,0}_0$ is shown to be isomorphic to some scale of Banach sequence spaces. This enables us to obtain some tightness criterion in these spaces. As an application, we prove the weak Hölder convergence of the convolution-smoothed empirical process of an i.i.d. sample $(X_1,...,X_n)$ under a natural assumption about the regularity of the marginal distribution function F of the sample. In particular, when F is Lipschitz, the best possible bound α<1/2 for the weak α-Hölder convergence of such processes is achieved.

Ograniczanie wyników

1 Applicationes Mathematicae

1 Colloquium Mathematicum

1 Bell M.

1 Hamadouche D.

1 Suquet C.

2 1999

Wyniki wyszukiwania

Tightness and π-character in centered spaces

Weak Hölder convergence of processes with application to the perturbed empirical process