The Limiting Eigenvalue Distribution of a Multivariate F Matrix

Let Xᵢⱼ ,Yᵢⱼ i,j = 1,2, ⋯, be i.i.d. $N(0,1)$ random variables and for positive integers $p,m,n $, let Xₚ = (Xᵢⱼ ) i = 1,2, ⋯ ,p; j = 1,2, ⋯ ,m, and Yₚ = (Yᵢⱼ ) i = 1,2, ⋯ ,p; j = 1,2, ⋯ ,n. Suppose further that p / m → y > 0 and p / n → y' ∈ (0,1/2) as p → ∞ . In [5], [6] it is shown that the empirical distribution function of the eigenvalues of (1 / m Xₚ XₚT )(1 / n Yₚ YₚT )- 1 converges i.p. as p → ∞ to a nonrandom d.f. In the present paper the limiting d.f. is derived.