Abstract
The main purpose of this paper is to study certain representations of sums of iid k-vector rv's as embeddings in k-dimensional Brownian motion by vectors of stopping times, in extension of Skorohod's scheme [20], and consequent error estimates for weak and strong invariance principles. In particular, letting k→∞ we embed the sample df in the Gaussian process with 2-dimensional time to which it has long been known to converge weakly. We discuss previous sample df embeddings, which have yielded related results; while some of our estimates are slight improvements, the emphasis here will be on the naturality of the embedding per se (although it will be indicated why it is probably far from the final word on the subject.
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