An Invariance Principle for Triangular Arrays

Let A n, i be a triangular array of sign-symmetric exchangeable random variables satisfying nE(A 2 n, i ) (longrightarrow)1, nE(A 4 n, i ) (longrightarrow)0, n 2 E(A 2 n, 1 A 2 n, 2) (longrightarrow)1. We show that (sigma)[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)= (sigma) [nt] i=1 A ii, then X n converges to Brownian motion. Similar results hold for the unitary group.