An invariance principle in L 2 [ 0 , 1 ] for non stationary ϕ -mixing sequences

Paulo Eduardo Oliveira; Charles Suquet

Commentationes Mathematicae Universitatis Carolinae (1995)

  • Volume: 36, Issue: 2, page 293-302
  • ISSN: 0010-2628

Abstract

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Invariance principle in L 2 ( 0 , 1 ) is studied using signed random measures. This approach to the problem uses an explicit isometry between L 2 ( 0 , 1 ) and a reproducing kernel Hilbert space giving a very convenient setting for the study of compactness and convergence of the sequence of Donsker functions. As an application, we prove a L 2 ( 0 , 1 ) version of the invariance principle in the case of ϕ -mixing random variables. Our result is not available in the D ( 0 , 1 ) -setting.

How to cite

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Oliveira, Paulo Eduardo, and Suquet, Charles. "An invariance principle in $L^2[0,1]$ for non stationary $\varphi $-mixing sequences." Commentationes Mathematicae Universitatis Carolinae 36.2 (1995): 293-302. <http://eudml.org/doc/247758>.

@article{Oliveira1995,
abstract = {Invariance principle in $L^2(0,1)$ is studied using signed random measures. This approach to the problem uses an explicit isometry between $L^2(0,1)$ and a reproducing kernel Hilbert space giving a very convenient setting for the study of compactness and convergence of the sequence of Donsker functions. As an application, we prove a $L^2(0,1)$ version of the invariance principle in the case of $\varphi $-mixing random variables. Our result is not available in the $D(0,1)$-setting.},
author = {Oliveira, Paulo Eduardo, Suquet, Charles},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {reproducing kernel Hilbert space; random measure; invariance principle; $\varphi $-mixing; invariance principle; -mixing; random measure; reproducing kernel Hilbert space},
language = {eng},
number = {2},
pages = {293-302},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {An invariance principle in $L^2[0,1]$ for non stationary $\varphi $-mixing sequences},
url = {http://eudml.org/doc/247758},
volume = {36},
year = {1995},
}

TY - JOUR
AU - Oliveira, Paulo Eduardo
AU - Suquet, Charles
TI - An invariance principle in $L^2[0,1]$ for non stationary $\varphi $-mixing sequences
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1995
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 36
IS - 2
SP - 293
EP - 302
AB - Invariance principle in $L^2(0,1)$ is studied using signed random measures. This approach to the problem uses an explicit isometry between $L^2(0,1)$ and a reproducing kernel Hilbert space giving a very convenient setting for the study of compactness and convergence of the sequence of Donsker functions. As an application, we prove a $L^2(0,1)$ version of the invariance principle in the case of $\varphi $-mixing random variables. Our result is not available in the $D(0,1)$-setting.
LA - eng
KW - reproducing kernel Hilbert space; random measure; invariance principle; $\varphi $-mixing; invariance principle; -mixing; random measure; reproducing kernel Hilbert space
UR - http://eudml.org/doc/247758
ER -

References

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  1. Billingsley P., Convergence of probability measures, Wiley, 1968. Zbl0944.60003MR0233396
  2. Berlinet A., Espaces autoreproduisants et mesure empirique, méthodes splines en estimation fonctionnelle, Thèse 3 cycle, Lille, 1980. 
  3. Davydov Y., Convergence of distributions generated by stationary stochastic processes, Theory Probab. Appl. 13 (1968), 691-696. (1968) Zbl0181.44101
  4. Guilbart C., Étude des produits scalaires sur l'espace des mesures. Estimation par projection. Tests à noyaux, Thèse d'Etat, Lille, 1978. 
  5. Herrndorf N., The invariance principle for ϕ -mixing sequences, Z. Wahrsch. verw. Gebiete 63 (1983), 97-108. (1983) MR0699789
  6. Ibragimov I.A., Some limit theorems for stationary processes, Theory Probability Appl. 7 (1962), 349-382. (1962) Zbl0119.14204MR0148125
  7. Jacob P., Private communication, . 
  8. Oliveira P.E., Invariance principles in L 2 ( 0 , 1 ) , Comment. Math. Univ. Carolinae 31:2 (1990), 357-366. (1990) MR1077906
  9. Parthasarathy K.R., Probability Measures on Metric Spaces, Academic Press, 1967. MR0226684
  10. Peligrad M., An invariance principle for ϕ -mixing sequences, Ann. Probab. 13 (1985), 1304-1313. (1985) MR0806227
  11. Philipp W., Invariance principles for independent and weakly dependent random variables, Dependence in probability and statistics (Oberwolfach, 1985), 225-268, Progr. Probab. Statist., 11, Birkhauser Boston, Boston, MA, 1986. Zbl0614.60027MR0899992
  12. Samur J.D., On the invariance principle for stationary ϕ -mixing triangular arrays with infinitely divisible limits, Probab. Th. Rel. Fields 75 (1987), 245-259. (1987) MR0885465
  13. Suquet Ch., Une topologie pré-hilbertienne sur l'espace des mesures à signe bornées, Pub. Inst. Stat. Univ. Paris XXXV (1990), 51-77. (1990) MR1745002
  14. Suquet Ch., Relecture des critères de relative compacité d'une famille de probabilités sur un espace de Hilbert, Pub. IRMA 28 (1992), III, Lille. 
  15. Suquet Ch., Convergences stochastiques de suites de mesures aléatoires à signe considérées comme variables aléatoires hilbertiennes, Pub. Inst. Stat. Univ. Paris XXXVII 1-2 (1993), 71-99. (1993) MR1743969
  16. Utev S.A., On the central limit theorem for ϕ -mixing arrays of random variables, Theory Probab. Appl. 35:1 (1990), 131-139. (1990) MR1050059

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