The Stieltjes transform in (3)can also be interpreted as an expectation with respect to the measure (r) such that Since there is a one-to-one correspondence between the probability measure FA(a) and the Stieltjes nsform, convergence of the Stieltjes transform can be used to show the convergence of the probability measure Fa(r). Once this convergence has been established, the Stieltjes transform can be used to yield the density using the so-called Stieltjes-Perron inversion formula [6 d=亓如mmm4(x+i) (5) When studying the limiting distribution of large random matrices, the Stieltjes transform has proved to be particularly relevant because of its correspondence with the matrix resolvent. The trace of the matrix resolvent, MA(z), defined as MA(z)=(AN-2r)-I can be written as trMA(z)l where Ai for i= 1, 2, are the eigenvalues of AN. For any AN, MA(z) is a non-random quantity However, when AN is a random matrix ma(z)=lim =tr[MA(a) The Stieltjes transform and its resolvent form in(7) are intimately linked to the classical moment problem 6. This connection can be observed by noting that the integral in (3) can be expressed as an analytic multipole" series expansion about 2= oo such that 2k+1 k+1 k=0 k=1 where MA=adFA()is the Ath moment of r on the probability measure dFA(a). The analyticity of the Stieltjes transform about z oo expressed in( 8)is a consequence of our implicit assumption that the region of support for the limiting density dFa(a)is bounded i.e. limz-oo dF(a)=0 Incidentally, the n-transform introduced by Tulino and Verdi in 7 can be expressed in terms of m(z) nd permits a series expansion about 2=0 Given the relationship in(7), it is worth noting that the matrix moment MA is simply MA= lim xtr[AN]=/rkdFA() Equation(8), written as a multipole series, suggests that a way of computing the density would be te determine the moments of the random matrix as in(9), and then invert the Stieltjes transform using(5).For the famous semi-circular law, Wigner actually used a moment based approach in [8 to determine the density for the standard Wigner matrix though he did not explicitly invert the Stieltjes transform as we suggested e reader may imagine, such a moment based approach is not particularly useful for more genera classes of random matrices. We discuss a more relevant and practically useful Stieltjes transform based� � � � The Stieltjes transform in (3) can also be interpreted as an expectation with respect to the measure F A(x) such that 1 mA(z) = EX . (4) x − z Since there is a one-to-one correspondence between the probability measure F A(x) and the Stieltjes transform, convergence of the Stieltjes transform can be used to show the convergence of the probability measure F A(x). Once this convergence has been established, the Stieltjes transform can be used to yield the density using the so-called Stieltjes-Perron inversion formula [6] dF A(x) 1 = lim Im mA(x + iξ). (5) dx π ξ→0 When studying the limiting distribution of large random matrices, the Stieltjes transform has proved to be particularly relevant because of its correspondence with the matrix resolvent. The trace of the matrix resolvent, MA(z), defined as MA(z) = (AN − zI)−1 can be written as � N 1 tr[MA(z)] = (6) λi − z i=1 where λi for i = 1, 2, . . . , N are the eigenvalues of AN . For any AN , MA(z) is a non-random quantity. However, when AN is a large random matrix, 1 mA(z) = lim tr[MA(z)]. (7) N→∞ N The Stieltjes transform and its resolvent form in (7) are intimately linked to the classical moment problem [6]. This connection can be observed by noting that the integral in (3) can be expressed as an analytic “multipole” series expansion about z = ∞ such that � ∞ k ∞ � k mA(z) = zk+1 dF A(x) = � x dF A(x) � x zk+1 − − k=0 k=0 ∞ (8) 1 � Mk A = . zk+1 −z − k=1 where Mk A = xkdF A(x) is the kth moment of x on the probability measure dF A(x). The analyticity of the Stieltjes transform about z = ∞ expressed in (8) is a consequence of our implicit assumption that the region of support for the limiting density dF A(x) is bounded i.e. limx→∞ dF A(x) = 0. Incidentally, the η-transform introduced by Tulino and Verd`u in [7] can be expressed in terms of m(z) and permits a series expansion about z = 0. Given the relationship in (7), it is worth noting that the matrix moment MA is simply k MA = lim 1 tr[Ak N ] = x kdF A(x). (9) k N→∞ N Equation (8), written as a multipole series, suggests that a way of computing the density would be to determine the moments of the random matrix as in (9), and then invert the Stieltjes transform using (5). For the famous semi-circular law, Wigner actually used a moment based approach in [8] to determine the density for the standard Wigner matrix though he did not explicitly invert the Stieltjes transform as we suggested above, As the reader may imagine, such a moment based approach is not particularly useful for more general classes of random matrices. We discuss a more relevant and practically useful Stieltjes transform based approach next