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

Djamel Hamadouche; Charles Suquet

Applicationes Mathematicae (1999)

  • Volume: 26, Issue: 1, page 63-83
  • ISSN: 1233-7234

Abstract

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We consider stochastic processes as random elements in some spaces of Hölder functions vanishing at infinity. The corresponding scale of spaces C 0 α , 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.

How to cite

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Hamadouche, Djamel, and Suquet, Charles. "Weak Hölder convergence of processes with application to the perturbed empirical process." Applicationes Mathematicae 26.1 (1999): 63-83. <http://eudml.org/doc/219226>.

@article{Hamadouche1999,
abstract = {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.},
author = {Hamadouche, Djamel, Suquet, Charles},
journal = {Applicationes Mathematicae},
keywords = {triangular functions; Schauder decomposition; Hölder space; tightness; Brownian bridge; perturbed empirical process},
language = {eng},
number = {1},
pages = {63-83},
title = {Weak Hölder convergence of processes with application to the perturbed empirical process},
url = {http://eudml.org/doc/219226},
volume = {26},
year = {1999},
}

TY - JOUR
AU - Hamadouche, Djamel
AU - Suquet, Charles
TI - Weak Hölder convergence of processes with application to the perturbed empirical process
JO - Applicationes Mathematicae
PY - 1999
VL - 26
IS - 1
SP - 63
EP - 83
AB - 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.
LA - eng
KW - triangular functions; Schauder decomposition; Hölder space; tightness; Brownian bridge; perturbed empirical process
UR - http://eudml.org/doc/219226
ER -

References

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  13. [13] G. R. Shorack and J. A. Wellner, Empirical Processes with Applications to Statistics, Wiley, 1986. Zbl1170.62365
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  16. [16] C. Suquet, Tightness in Schauder decomposable Banach spaces, Translations of A.M.S., Proceedings of the St Petersburg Math. Soc., to appear. 

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