Invariance Principles in Probability for Triangular Arrays of B-Valued Random Vectors and Some Applications

If μₙ, ν are probability measures on a separable Banach space, jₙ → ∞ and μʲⁿₙ →w ν (so ν is necessarily infinitely divisible), then it is possible to construct two row-wise independent triangular arrays ₙⱼ\, ₙⱼ\ such that L(Xₙⱼ) = μₙ, L(Yₙⱼ) = ν1/jn and maxk ≤ jn \|Sₙₖ - Tₙₖ\|→P 0, where Sₙₖ and Tₙₖ are the respective partial row sums. Several refinements are proved. These results are applied to establish the weak convergence of the distributions of certain functionals of the partial row sums, improving well-known results of Skorohod. As concrete applications, we prove an arc-sine law for triangular arrays generalizing the Erdos-Kac law and an arc-sine law for strictly stable processes generalizing P. Levy's law for Brownian Motion.