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
    • Department of MathematicsLouisiana State University
  • Joel Zinn
    • Department of Statistics and ProbabilityMichigan State University

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.

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© Springer-Verlag 1983