Abstract
Let A n, i be a triangular array of sign-symmetric exchangeable random variables satisfying nE(A 2 n, i )→1, nE(A 4 n, i )→0, n 2 E(A 2 n, 1 A 2 n, 2)→1. We show that ∑[nt] i=1 A ni, 0≤t≤1, converges to Brownian motion. This is applied to show that if A is chosen from the uniform distribution on the orthogonal group O n and X n(t)=∑[nt] i=1 A ii, then X n converges to Brownian motion. Similar results hold for the unitary group.
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D'Aristotile, A. An Invariance Principle for Triangular Arrays. Journal of Theoretical Probability 13, 327–341 (2000). https://doi.org/10.1023/A:1007801726073
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DOI: https://doi.org/10.1023/A:1007801726073