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

Abstract

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∥Sn∥pwhich 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.