© Hindawi Publishing Corp. ON CHUNG-TEICHER TYPE STRONG LAW FOR ARRAYS OF VECTOR-VALUED RANDOM VARIABLES

We study the equivalence between the weak and strong laws of large numbers for arrays of row-wise independent random elements with values in a Banach space . The conditions under which this equivalence holds are of the Chung or Chung-Teicher types. These con-ditions are expressed in terms of convergence of specific series and o(1) requirements on specific weighted row-wise sums. Moreover, there are not any conditions assumed on the geometry of the underlying Banach space. 2000 Mathematics Subject Classification: 60F15, 60B12. Let (Ω,,P) be a probability space and let be a real separable Banach space with norm ‖·‖. A strongly measurable transformation from Ω to is said to be a -valued random variable or a random element. If E‖X‖<∞, then the expected value is defined by the Bochner integral. Let {Xn, n ≥ 1} be a sequence of -valued random variables. Then {Xn, n ≥ 1} is said to obey the strong law of large numbers (SLLN) if there exist sequences of real numbers {an, n ≥ 1} and {bn, n ≥ 1} such that n∑ j=1 aj Xj−bj → 0 a.s., n →∞. (1) Sufficient conditions for SLLN use very often the geometry of a Banach space, that is, they assume that is a special-type space, for instance is of Rademacher type p