Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete

, Volume 62, Issue 3, pp 323-354

First online:

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

  • Evarist GinéAffiliated withDepartment of Mathematics, Louisiana State University
  • , Joel ZinnAffiliated withDepartment of Statistics and Probability, Michigan State University

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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∥S npwhich in l pspaces, 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 l p, p<2. Several examples and counterexamples are provided.