On strong convergence in L 1 for a sequence of bochner integrable random vectors

Luca Pratelli

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1992)

  • Volume: 3, Issue: 4, page 299-306
  • ISSN: 1120-6330

Abstract

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The problem of finding simple additional conditions, for a weakly convergent sequence in L 1 , which would suffice to imply strong convergence has been widely studied in recent years. In this Note we study this problem for Banach valued random vectors, by replacing weak convergence with a less restrictive assumption. Moreover, all the additional conditions we consider are also necessary for strong convergence, and they depend only on marginal distributions.

How to cite

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Pratelli, Luca. "Sur la convergence en moyenne pour des vecteurs aléatoires intégrables au sens de Bochner." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 3.4 (1992): 299-306. <http://eudml.org/doc/244174>.

@article{Pratelli1992,
author = {Pratelli, Luca},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Strong convergence in L1; Strict convexity; Uniform integrability; Convergence in distribution; Bochner integrability; Banach-space-valued random vectors; Bochner-integrable random vectors; convergence in mean},
language = {fre},
month = {12},
number = {4},
pages = {299-306},
publisher = {Accademia Nazionale dei Lincei},
title = {Sur la convergence en moyenne pour des vecteurs aléatoires intégrables au sens de Bochner},
url = {http://eudml.org/doc/244174},
volume = {3},
year = {1992},
}

TY - JOUR
AU - Pratelli, Luca
TI - Sur la convergence en moyenne pour des vecteurs aléatoires intégrables au sens de Bochner
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1992/12//
PB - Accademia Nazionale dei Lincei
VL - 3
IS - 4
SP - 299
EP - 306
LA - fre
KW - Strong convergence in L1; Strict convexity; Uniform integrability; Convergence in distribution; Bochner integrability; Banach-space-valued random vectors; Bochner-integrable random vectors; convergence in mean
UR - http://eudml.org/doc/244174
ER -

References

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  2. BILINGSLEY, P., Convergence of probability measures. New York1968. Zbl0172.21201
  3. DELLACHERIE, C. - MEYER, P. A., Probabilités et potentiel. Chap. I-IV, Paris1975. Zbl0323.60039MR488194
  4. DIESTEL, J. - UHL, J., Vector measures. Math. Surveys, 15, AMS, 1978. Zbl0369.46039MR453964
  5. PRATELLI, L., Sur une caractérisation de la convergence dans L 1 . Application aux quasimartingales. Sémin. Prob., à paraître. Zbl0766.60041
  6. RZEZUCHOWSKI, T., Impact of dentability on weak convergence in L 1 . Boll. UMI, (7) 6-A, 1992, 71-80. Zbl0762.40002MR1164737
  7. VALADIER, M., Différents cas où, grâce à une propriété d'extrémalité, une suite de fonctions intégrables faiblement convergente converge fortement. Exposé n. 5, 1989, Sémin. d'Analyse Convexe, Montpellier. Zbl0734.49004MR1061750
  8. VISINTIN, A., Strong convergence results related to strict convexity. Comm. Partial Diff. Equations, 9(5), 1984, 439-466. Zbl0545.49019MR741216DOI10.1080/03605308408820337

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