Figure 3: The matrices Bn and CN when n <N Multiplying both sides of(25) by c, we get cmBN(2)=-(1-c)-+mcn(2) which upon rearranging the terms is precisely(22). Equation(22)is an eract expression relating the Stieltjes transforms of Cn and BN for all n and N. As N- 00, the Marcenko-Pastur theorem states that mBN(2)mB(a) which implies that the limiting Stieltjes transform mc(a), for Cn can be written in terms of mB(z) using(22) 1 +cmB(a) For our purpose of getting the canonical equation for mc(a) using the canonical equation for mB(a),it is more useful to rearrange the terms in(27)and express mB(a) can be written in terms of mc(z). This elationship is simply mB(2) (a-) 2+ -mc(a) Hence, to obtain the canonical equation for mc(a) we simply have to substitute the expression for mB(z) in(28)into(13). With some fairly straightforward algebra, that we shall omit here, it can be verified that mc(a) is the solution to the canonical equation dH(r) {(1-c-czmc(2)}7-z (29) Incidentally, (29)was first derived by Silverstein in[10). He noted that the eigenvalues of +Tl Xnx: Tl/ were the same as those of xXnXnTn so that(29)was the canonical equation for this class of matrices as well. Additionally, in their proof of the Marcenko-Pastur theorem in 11, Bai and Silverstein dropped the restriction on Tn being diagonal so that Tn could be any matrix whose e d f. FIn-H. As the reader may appreciate, this broadens the class of matrices for which this theorem may be applied 3.2 The(generalized) Wishart matrix Revisiting the previous example, when Tn=l, Cn=NXnXn. This matrix Cn is the generalized version of he famous Wishart matrix ensemble first studied by Wishart in 1928 12. In physics literature, the Wishart matrix is also referred to as the Laguerre Ensemble [13. Strictly speaking, Cn is referred to as a Wishart matrix only when the elements of Xn are i.i. d. Gaussian random variables. The canonical equation for C in(29) beco hich upon rearranging yields the Stieltjes polynomia c zm)-2 n三 czm2-(1-c-2)m+1=0. (31)� � � n N n n n T n n 1/2 n Xn T 1/2 n X∗ N n n n nn n n N X∗ N n T 1/2 n T 1/2 n Xn 1 1 N On n N YN × YN × O∗ ∗ = = Cn BN Figure 3: The matrices Bn and CN when n < N. Multiplying both sides of (25) by c, we get 1 c mBN (z) = −(1 − c) + mCn (z) (26) z which upon rearranging the terms is precisely (22). Equation (22) is an exact expression relating the Stieltjes transforms of Cn and BN for all n and N. As N → ∞, the Marˇcenko-Pastur theorem states that mBN (z) → mB(z) which implies that the limiting Stieltjes transform mC(z), for Cn can be written in terms of mB(z) using (22) as mC(z) = c − 1 − + c mB(z). (27) z For our purpose of getting the canonical equation for mC (z) using the canonical equation for mB(z), it is more useful to rearrange the terms in (27) and express mB(z) can be written in terms of mC (z). This relationship is simply 1 1 1 mB(z) = − c − 1 z + mC (z). (28) c Hence, to obtain the canonical equation for mC(z) we simply have to substitute the expression for mB(z) in (28) into (13). With some fairly straightforward algebra, that we shall omit here, it can be verified that mC (z) is the solution to the canonical equation dH(τ) mC(z) = . (29) {(1 − c − c z mC (z)}τ − z 1/2 ∗ 1/2 Incidentally, (29) was first derived by Silverstein in [10]. He noted that the eigenvalues of 1 N Tn XnXnTn ∗ were the same as those of 1 XnXnTn so that (29) was the canonical equation for this class of matrices as N well. Additionally, in their proof of the Marˇcenko-Pastur theorem in [11], Bai and Silverstein dropped the restriction on Tn being diagonal so that Tn could be any matrix whose e.d.f. F Tn → H. As the reader may appreciate, this broadens the class of matrices for which this theorem may be applied. 3.2 The (generalized) Wishart matrix 1 ∗ Revisiting the previous example, when Tn = I, Cn = N XnX . This matrix Cn n is the generalized version of the famous Wishart matrix ensemble first studied by Wishart in 1928 [12]. In physics literature, the Wishart matrix is also referred to as the Laguerre Ensemble [13]. Strictly speaking, Cn is referred to as a Wishart matrix only when the elements of Xn are i.i.d. Gaussian random variables. The canonical equation for Cn in (29) becomes 1 m = (30) (1 − c − c z m) − z which upon rearranging yields the Stieltjes polynomial 2 c z m − (1 − c − z)m + 1 = 0. (31)