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Let X, X1, X2,... be a sequence of nondegenerate i.i.d. random variables, let μ = {μni : n ∈ ℕ+, i = 1, ..., n} be a triangular array of possibly random probabilities on the interval [0, 1], and let F be a class of functions with bounded q-variation on [0, 1] for some q ∈ [1, 2). We prove the asymptotic normality uniformly over F of self-normalized weighted sums ∑ i = 1 n X i μ ni when μ is the array of point measures, uniform probabilities, and their random versions. Also, we prove a weak invariance principle in the Banach space of functions of bounded p-variation with p > 2 for partial-sum processes, polygonal processes, and their adaptive versions. [ABSTRACT FROM AUTHOR]