Summary
We consider a minimal form of the usual conditions for the dependent central limit theorem and invariance principle for “near martingales”. We show that these conditions imply convergence to Brownian motion in a way that is slightly stronger than weak convergence in D[0,∞). On the other hand, if a sequence of processes with paths in D[0,∞) converges to Brownian motion in this way, then we can always find a sequence of partitions of the time axis that is such that these conditions hold for the corresponding array of increments.
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Helland, I.S. On weak convergence to Brownian motion. Z. Wahrscheinlichkeitstheorie verw Gebiete 52, 251–265 (1980). https://doi.org/10.1007/BF00538890
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DOI: https://doi.org/10.1007/BF00538890