The convergence of the empirical distribution of canonical correlation coefficients

Abstract

Suppose that {X jk,j=1,⋯,p 1;k=1,⋯,n} are independent and identically distributed (i.i.d) real random variables with EX 11=0 and EX 211=1, and that {Y jk,j=1,⋯,p 2;k=1,⋯,n} are i.i.d real random variables with EY 11=0 and EY 211=1, and that {X jk,j=1,⋯,p 1;k=1,⋯,n} are independent of {Y jk,j=1,⋯,p 2;k=1,⋯,n}. This paper investigates the canonical correlation coefficients r 1≥r 2≥⋯≥r p1, whose squares λ 1=r 21,λ 2=r 22,⋯,λ p1=r 2p1 are the eigenvalues of the matrix S xy=A -1xA xyA -1yA Txy, where and x k=(X 1k,⋯,X p1k) T, y k=(Y 1k,⋯,Y p2k) T, k=1,⋯,n. When p 1→∞, p 2→∞ and n→∞ with p1n→c 1, p2n→c 2, c 1, c 2 ∈ (0,1), it is proved that the empirical distribution of r 1, r 2,⋯, r p1 converges, with probability one, to a fixed distribution under the finite second moment condition.