Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete

, Volume 67, Issue 3, pp 349–362

Une extension de la loi des grands nombres de Prohorov

Authors

  • Bernard Heinkel
    • Département de Mathématique
Article

DOI: 10.1007/BF00535009

Cite this article as:
Heinkel, B. Z. Wahrscheinlichkeitstheorie verw Gebiete (1984) 67: 349. doi:10.1007/BF00535009

Summary

The Prohorov law of large numbers is extended to random variables taking their values in a 2-uniformly smooth Banach space (B, ∥ ∥). In our result the classical assumption of the scalar case is replaced by the following one:
$$\forall \varepsilon > 0,\sum\limits_{n \in \mathbb{N}} {\exp } ( - \varepsilon /\Lambda {\text{(}}n{\text{)}}) < + \infty ,$$
where:
$$\Lambda {\text{(}}n{\text{)}} = 2^{ - 2n} (\mathop \sum \limits_{2^n + 1 \leqq j \leqq 2^{n + 1} } \sup (Ef^2 (X_j ),\parallel f\parallel _{B'} \leqq 1)).$$

Copyright information

© Springer-Verlag 1984