The convergence of the empirical distribution of canonical correlation coefficients - Yang, Yanrong (2013)
Suppose that {Xij, j = 1,..., p1; k = 1,...,n} are independent and identically distributed (i.i.d) real random variables with EX11 = 0 and EX112 = 1, and that {Yjk, j = 1,..., p2; k = 1,..., n} are i.i.d real random variables with EY11 = 0 and EY112 = 1, and that {Xjk, j = 1,..., p1; k = 1,..., n} are independent of {Yjk, j = 1,..., p2; k = 1,..., n}. This paper investigates the canonical correlation coefficients r1 ≥ r2 ≥ ... ≥ rp1, whose squares λ1 = r12, λ2 = r22,..., λp1 = rp12 are the eigenvalues of the matrix Sxy = Ax-1AxyAy-1AxyT, where Ax = 1/n∑xkxkT, Ay = 1/n∑ykykT, Axy = 1/n∑xkykT, and xk = (X1k,..., Xp1k)T, yk = (Y1k,..., Yp2k)T, k = 1,..., n. When p1 → ∞, p2 → ∞ and n → ∞ with p1/n → c1, p2/n → c2, c1, c2 ∈ (0,1), it is proved that the empirical distribution of r1, r2,..., rp1 converges, with probability one, to a fixed distribution under the finite second moment condition.