A weak law for normed weighted sums of random elements in rademacher type p banach spaces

For weighted sums [Sigma]j = 1najVj of independent random elements {Vn, n >= 1} in real separable, Rademacher type p (1 <= p <= 2) Banach spaces, a general weak law of large numbers of the form ([Sigma]j = 1najVj - vn)/bn -->p 0 is established, where {vn, n >= 1} and bn --> [infinity] are suitable sequences. It is assumed that {Vn, n >= 1} is stochastically dominated by a random element V, and the hypotheses involve both the behavior of the tail of the distribution of V and the growth behaviors of the constants {an, n >= 1} and {bn, n >= 1}. No assumption is made concerning the existence of expected values or absolute moments of the {Vn, n >- 1}.