Strong Law of Large Numbers of Pettis-Integrable Multifunctions

Using reversed martingale techniques, we prove the strong law of large numbres for independent Pettis-integrable multifunctions with convex weakly compact values in a Banach space. The Mosco convergence of reversed Pettis-integrable martingale of the form (EBnX)n≥1, where (Bn)n≥1 is a decreasing sequence of the sub σ-algebra of F is provided.

Tail sums of convergent series of independent random variables

Let X1, X2, … be a sequence of independent random variables such that, for each n ≥ 1, EXn = 0 and and assume that then converges almost surely as N → ∞. Let and let Fn(x) denote the distribution function of Xn. Loynes (2) observed that the sequence {Sn} is a reversed martingale, and applied his central limit theorem to it: however, stronger results are obtainable, in precise duality with the classical theory of partial sums of independent random variables. These results describe the fluctuations of the sequence {Sn}, and hence the way in which converges to its limit.