Figure 5: The density of Cn with c=n/N=0.1 and dH()=0.68(T-1)+0.48(T-3 The Wishart matrix W(c) has been studied exhaustively by statisticians and engineers. For infinite Wishart matrices, (32)captures the limiting density and the extreme eigenvalues i.e. the region of support The limiting moments are given by (34). As it was for the asymptotic Wigner matrix, the limiting density of the asymptotic Wishart matrix did not depend on whether the elements of Xn were real or complex. Similarly, as it was for the Gaussian ensembles, the analytical behavior of the finite Wishart matrices did indeed depend on whether the elements were real or complex. Nonetheless, the limiting moment and eigenvalue behavior could be inferred from the behavior of the finite (real or complex) Wishart matrix counterpart. The reader is directed towards some of the representative literature and the references therein on the distribution of the smallest [17, largest [18, 19, 20, 21, sorted [19, 22, 23, unsorted eigenvalues [19, 22, 23, and condition numbers 22, 23 of the Wishart matrix that invoke this link between the finite and infinite Wishart matrix ensembles. We will now discuss sample covariance matrices for which the behavior of the limiting density can be best, if not solely, analyzed analytically using the Marcenko-Pastur theore 3.3 Additional examples of sample covariance matrices Suppose dH()=pS(T-A1)+(1-p)S(T-A2)i.e. Tn has an atomic mass of weight p at A1 and another atomic mass of weight(1-p)at A2. The canonical equation in(29)becomes A1(1-c-cm2)-2 A2(1 (35) which upon rearranging yields the Stieltjes polynomial A1c2m32A2+(-2A1cz+A1c2+2h1c2A2+A2c2)m2 +(A1A2+ A2cz +pA2cz-A12+1c2A2+22-A22+2A1Cz-pAlc2-2A1A2c)m (P2+z-p1c+入1c-A1+pA2c+pA1)=0.(36) It can be readily verified that if p=l and A,=1, then C,=1Tl/X,x* Tl/ is simply the Wishart matrix e discussed above. Though it might not seem so from a cursory look, for p= 1 and A1= 1,(36)can be shown after some elementary factorization to simplify to(31). Since(36) is a third degree polynomial in it can conceivably ved analytically using Cardano's formula For general c, p, A and A2 this is cumbersome and cannot be solved analytically as a function of z and c for arbitrary values of p, A1, and A2. However, for specific values of c, p, A1, and A2 we can numerically solve� 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 dFC/dx 0.1 0 −1 0 1 2 3 4 5 6 x Figure 5: The density of Cn with c = n/N = 0.1 and dH(τ) = 0.6 δ(τ − 1) + 0.4 δ(τ − 3). The Wishart matrix {W(c)} has been studied exhaustively by statisticians and engineers. For infinite Wishart matrices, (32) captures the limiting density and the extreme eigenvalues i.e. the region of support. The limiting moments are given by (34). As it was for the asymptotic Wigner matrix, the limiting density of the asymptotic Wishart matrix did not depend on whether the elements of Xn were real or complex. Similarly, as it was for the Gaussian ensembles, the analytical behavior of the finite Wishart matrices did indeed depend on whether the elements were real or complex. Nonetheless, the limiting moment and eigenvalue behavior could be inferred from the behavior of the finite (real or complex) Wishart matrix counterpart. The reader is directed towards some of the representative literature and the references therein on the distribution of the smallest [17], largest [18, 19, 20, 21], sorted [19, 22, 23], unsorted eigenvalues [19, 22, 23], and condition numbers [22, 23] of the Wishart matrix that invoke this link between the finite and infinite Wishart matrix ensembles. We will now discuss sample covariance matrices for which the behavior of the limiting density can be best, if not solely, analyzed analytically using the Marˇcenko-Pastur theorem. 3.3 Additional examples of sample covariance matrices Suppose dH(τ) = p δ(τ − λ1) + (1 − p) δ(τ − λ2) i.e. Tn has an atomic mass of weight p at λ1 and another atomic mass of weight (1 − p) at λ2. The canonical equation in (29) becomes p 1 − p m = + (35) λ1(1 − c − cmz) − z λ2(1 − c − cmz) − z which upon rearranging yields the Stieltjes polynomial 2� 2 λ1 c 2 m 3 z 2λ2 + � −2λ1λ2cz + λ1cz 2 + 2λ1c 2λ2z + λ2cz m 2 + λ1λ2 + λ2cz + pλ2cz − λ1z + λ1c 2λ2 + z − λ2z + 2λ1cz − pλ1cz − 2λ1λ2 c � m − (pλ2 + z − pλ1c + λ1c − λ1 + pλ2c + pλ1) = 0. (36) 1 1/2 ∗ 1/2 It can be readily verified that if p = 1 and λ1 = 1, then Cn = N Tn XnXnTn is simply the Wishart matrix we discussed above. Though it might not seem so from a cursory look, for p = 1 and λ1 = 1, (36) can be shown after some elementary factorization to simplify to (31). Since (36) is a third degree polynomial in m it can conceivably be solved analytically using Cardano’s formula. For general c, p, λ1 and λ2 this is cumbersome and cannot be solved analytically as a function of z and c for arbitrary values of p, λ1, and λ2. However, for specific values of c, p, λ1, and λ2 we can numerically solve