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

AbstractFor weighted sums Σj = 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 (Σj = 1najVj − vn)bn →p 0 is established, where {vn, n ≥ 1} and bn → ∞ 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}