Some Strong Laws of Large Numbers for Banach Space Valued Summands Irrespective of Their Joint Distributions

A Random Set Extension of a Strong Law of Large Numbers of Cantrell and Rosalsky

“…Our purpose is to point out an extension to random sets of a strong law of large numbers (SLLN), recently reported by Cantrell and Rosalsky [2], in which almost sure convergence of the type…”

On Cantrell–Rosalsky’s strong laws of large numbers

“…Starting from a simple result of convergence a.s. for nonnegative random variables (Theorem 1), we establish the main result in (Cantrell and Rosalsky, 2003) as a corollary, and we show that some formal simplifications are even possible in the statement of hypothesis.…”

“…Next result is in the spirit of Theorem 2 in (Cantrell and Rosalsky, 2003), with some modifications in hypothesis: we suppress condition…”

“…Recently, Terán (2005) proved an extension to random sets of Theorem 1 in (Cantrell and Rosalsky, 2003). We are going to prove that it is also possible to extend our Corollary 2 to the setting of random sets in the same sense.…”

“…Taylor, 1978). Cantrell and Rosalsky (2003) established SLLNs for normed and centered sums of random elements {V n , n ≥ 1} irrespective of their joint distributions and without any condition of independence or pairwise independence on them. No geometric condition was imposed on the Banach space X and conditions on the random elements involved only their marginal distributions.…”