3 Stieltjes transform based approach Instead of obtaining the Stieltjes transform directly, the so-called Stieltjes transform approach relies instead on finding a canonical equation that the Stieltjes transform satisfies. The Marcenko-Pastur theorem 9 was the first and remains most famous example of such an approach. We include a statement of its theorem in the form found in literature. We encourage you to write this theorem, as an exericse, in a simpler manner. Theorem 1(The Marcenko-Pastur Theorem ). Consider an N X N matriT, BN. Assume that 1. Xn is an nxN matriz such that the matriz elements Xn are independent identically distributed (i i.d. opler random variables with mean zero and variance I i.e. XmEC, EIXml=0 and Ell Xm=1 2.n=n(N)uihn/N→c>0asN→∞ 3. Tn= diag(T, T2, .. Tn) where Ti E R, and the e d f. of Ti,..., Tn converges almost surely in distribution to a probability distribution function H(T)as N-o0 4. BN=AN+NXnTnXn, where AN is a Hermitian NXN matriz for which FAN converges vaguely to A almost surely, A being a possibly defective(i.e. with discontinuities) nonrandom distribution function Xn, Tn, and AN are independent Then, almost surely, FoN converges vaguely, almost surely, as N-oo to a nonrandom d f. F whose Stieltjes transform m(2), zE C, satisfies the canonical equation dhl m(2)=mA(z- 1+m(z) (10) We now illustrate the use of this theorem with a representative example. This example will help highlight issues that will be of pedagogical interest throughout this semester Suppose AN=0ie. BN=NXnTnXn. The Stieltjes transform of AN, by the definition in(3), is then Hence, using the Marcenko-Pastur theorem as expressed in(10), the Stieltjes transform m(z)of BN is given Rearranging the terms in this equation and using m instead of m(z) for notational convenience, we get dH(T) (13) Equation(13)expresses the dependence between the Stieltjes transform variable m and probability space variable z. Such a dependence, expressed explicitly in terms of dH(T), will be referred to throughout this paper as a canonical equation. Equation(13)can also be interpreted as the expression for the functional inverse of m(z) To determine the density of BN by using the inversion formula in(5) we need to first solve(13)for m(z) In order to obtain an equation in m and z we need to first know dH(T)in(13). In theory, dH(r) could be any density that satisfies the conditions of the Marcenko-Pastur theorem. However, as we shall shortly recognize, for an arbitrary distribution, it might not be possible to obtain an analytical or even an easy numerical solution for the density On the other hand, for some specific distributions of dH(T), it will indeed be possible to analytically obtain the density We consider one such distribution below L ppose T=I i.e. the diagonal elements of Tn are non-random with d.f. dH(T)=8(T-1).Equation (14)� � � � 3 Stieltjes transform based approach Instead of obtaining the Stieltjes transform directly, the so-called Stieltjes transform approach relies instead on finding a canonical equation that the Stieltjes transform satisfies. The Marˇcenko-Pastur theorem [9] was the first and remains most famous example of such an approach. We include a statement of its theorem in the form found in literature. We encourage you to write this theorem, as an exericse, in a simpler manner. Theorem 1 (The Marˇcenko-Pastur Theorem ). Consider an N × N matrix, BN . Assume that 1. Xn is an n×N matrix such that the matrix elements Xn ij are independent identically distributed (i.i.d.) complex random variables with mean zero and variance 1 i.e. Xn ij ] = 0 and E[kXn = 1. ij ∈ C, E[Xn ijk2] 2. n = n(N) with n/N → c > 0 as N → ∞. 3. Tn = diag(τ1 n, τ2 n, . . . , τ n n } converges almost surely in n ) where τi n ∈ R, and the e.d.f. of {τ1 n, . . . , τ n distribution to a probability distribution function H(τ) as N → ∞. 1 4. BN = AN + N X∗TnXn, where AN is a Hermitian N ×N matrix for which F AN n converges vaguely to A almost surely, A being a possibly defective (i.e. with discontinuities) nonrandom distribution function. 5. Xn, Tn, and AN are independent. Then, almost surely, F BN converges vaguely, almost surely, as N → ∞ to a nonrandom d.f. F B whose Stieltjes transform m(z), z ∈ C, satisfies the canonical equation τ dH(τ) m(z) = mA z − c (10) 1 + τ m(z) We now illustrate the use of this theorem with a representative example. This example will help us highlight issues that will be of pedagogical interest throughout this semester. 1 Suppose AN = 0 i.e. BN = N X∗TnXn. The Stieltjes transform of AN , by the definition in (3), is then n simply 1 1 mA(z) = = . (11) 0 − z −z Hence, using the Marˇcenko-Pastur theorem as expressed in (10), the Stieltjes transform m(z) of BN is given by 1 m(z) = � . (12) τ dH(τ) −z − c 1+τm(z) Rearranging the terms in this equation and using m instead of m(z) for notational convenience, we get 1 τ dH(τ) z = + − c . (13) m 1 + τm Equation (13) expresses the dependence between the Stieltjes transform variable m and probability space variable z. Such a dependence, expressed explicitly in terms of dH(τ), will be referred to throughout this paper as a canonical equation. Equation (13) can also be interpreted as the expression for the functional inverse of m(z). To determine the density of BN by using the inversion formula in (5) we need to first solve (13) for m(z). In order to obtain an equation in m and z we need to first know dH(τ) in (13). In theory, dH(τ) could be any density that satisfies the conditions of the Marˇcenko-Pastur theorem. However, as we shall shortly recognize, for an arbitrary distribution, it might not be possible to obtain an analytical or even an easy numerical solution for the density On the other hand, for some specific distributions of dH(τ), it will indeed be possible to analytically obtain the density We consider one such distribution below. Suppose Tn = I i.e. the diagonal elements of Tn are non-random with d.f. dH(τ) = δ(τ − 1). Equation (13) then becomes 1 c z = + − . (14) m 1 + m