Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete

, Volume 62, Issue 3, pp 323–354

Central limit theorems and weak laws of large numbers in certain banach spaces

  • Evarist Giné
  • Joel Zinn

DOI: 10.1007/BF00535258

Cite this article as:
Giné, E. & Zinn, J. Z. Wahrscheinlichkeitstheorie verw Gebiete (1983) 62: 323. doi:10.1007/BF00535258


For B a type 2 Banach lattice, we obtain a relationship between the central limit theorem in B and the weak law of large numbers (for the sum of the squares of the random vectors) in another Banach lattice B(2). We then obtain some two-sided estimates for E∥Snpwhich in lpspaces, 1≦p<∞, give n.a.s.c. for the weak law of large numbers. As a consequence of these estimates we also solve the domain of attraction problem in lp, p<2. Several examples and counterexamples are provided.

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • Evarist Giné
    • 1
  • Joel Zinn
    • 2
  1. 1.Department of MathematicsLouisiana State UniversityBaton RougeUSA
  2. 2.Department of Statistics and ProbabilityMichigan State UniversityEast LansingUSA
  3. 3.Department of MathematicsTexas A & M UniversityCollege StationUSA