The convergence of the empirical distribution of canonical correlation coefficients

Suppose that ⱼₖ, 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 ⱼₖ, j=1,⋯,p₂; k=1,⋯,n\ are i.i.d real random variables with EY₁₁=0 and EY₁₁²=1, and that ⱼₖ, j=1,⋯,p₁; k=1,⋯,n\ are independent of ⱼₖ, 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{equation*}Sxy=A_x⁻¹AxyA_y⁻¹AxyT,{equation*}where{equation*}A_x=1/n∑ⁿₖ₌₁x_kx_kT,A_y=1/n∑ⁿₖ₌₁y_ky_kT,Axy=1/n∑ⁿₖ₌₁x_ky_kT,{equation*}and{equation*}x_k=(X₁ₖ,⋯,Xp_1k)T,y_k=(Y₁ₖ,⋯,Yp_2k)T,\ k=1,⋯,n.{equation*} 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.