c=0.01 Figure 4: The density of the(generalized)Wishart matrix Wn(c) for different values of c Once again,(31) is a second degree polynomial in m whose coefficients are polynomials in 2. Hence, as before, (31)can be solved analytically to yield a solution for m in terms of z. The inversion formula in(5 can then be used to obtain the limiting density for Cn. This is simply max[0, r-H where, as before, b+=(1+vc)2. As(32)suggests, the limiting density for Cn depends only on the parameter C. Hence, for the remainder of this paper, we will use the notation W(c) to denote the(generalized )Wishart matrix ensemble defined as W(c)=NXnXn with c=n/N>0 Figure 4 plots the density in(32)for different values of c. From(32)the reader may notice that as c-0 i.e. for a fixed n as N- 00, both the largest and the smallest eigenvalue, b+ and b_- respectively, approach 1. This validates our intuition about the behavior of w(c) for a fixed n as N- o0. Additionally, from Figure 4, the reader may notice that the density gets increasingly more symmetrical about z l as the value of c decreases. For c=0.01, the density is almost perfectly symmetrical about z 1. The reader may note that with an appropriate scaling and translation, the density of w(c) as c-0 could be made to resemble the semi-circular distribution. In fact, in [14 Jonsson used this observation and the correspondence between the distribution in(32)and the generalized B-distribution to infer the moments of Wn(c) from the even moments of the Wigner matrix which are incidentally the Catalan numbers denoted by Ck for an integer k More recently, Dumitriu recognized [15] that these moments could be written in terms of the(k, r)Narayana umbers [16 defined as 1/k)/k-1 (33) so that the individual moments may be obtained from the moment generating function M(a=N=>()(7) for which, it may be noted that Mp(1)=Ck=M%k are also the even moments of the standard Wigner matrix en discussing infinite(generalized) Wishart matrices we will simply to W(e)as Wishart matrix e elements of Xn are i.i. d. but not Gaussian. We will occasionally add a subscript such as Wi(c) differentiate between realizations of the ensemble W(c). When discussing finite Wishart matrices, we will implicitly ume that the elements of X are i.i.d. Gaussian. Its use in either manner will be obvious from the context1 2 3 � � � � �� � � � 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1.5 2.5 3.5 dFC/dx c=1 c=0.5 c=0.1 c=0.01 x Figure 4: The density of the (generalized) Wishart matrix {Wn(c)} for different values of c. Once again, (31) is a second degree polynomial in m whose coefficients are polynomials in z. Hence, as before, (31) can be solved analytically to yield a solution for m in terms of z. The inversion formula in (5) can then be used to obtain the limiting density for Cn. This is simply dF C (x) 1 = max 0, 1 − δ(x) + (x − b−)(b+ − x) I[ b− ,b+] (32) dx c 2πxc where, as before, b± = (1±√c)2 . As (32) suggests, the limiting density for Cn depends only on the parameter c. Hence, for the remainder of this paper, we will use the notation {W(c)} to denote the (generalized) Wishart matrix ensemble 4 1 ∗ defined as W(c) = N XnXn with c = n/N > 0. Figure 4 plots the density in (32) for different values of c. From (32) the reader may notice that as c → i.e. for a fixed n as N → ∞, both the largest and the smallest eigenvalue, b+ and b− respectively, approach 1. This validates our intuition about the behavior of W(c) for a fixed n as N → ∞. Additionally, from Figure 4, the reader may notice that the density gets increasingly more symmetrical about x = 1 as the value of c decreases. For c = 0.01, the density is almost perfectly symmetrical about x = 1. The reader may note that with an appropriate scaling and translation, the density of {W(c)} as c → 0 could be made to resemble the semi-circular distribution. In fact, in [14] Jonsson used this observation and the correspondence between the distribution in (32) and the generalized β-distribution to infer the moments of Wn(c) from the even moments of the Wigner matrix which are incidentally the Catalan numbers denoted by Ck for an integer k. More recently, Dumitriu recognized [15] that these moments could be written in terms of the (k, r) Narayana numbers [16] defined as Nk,r = 1 k k − 1 (33) r + 1 r r so that the individual moments may be obtained from the moment generating function k−1 k−1 � �� � M r k W (c) = c rNk,r = c k k − 1 (34) r r r=0 r=0 for which, it may be noted that Mk W (1) = Ck = M2 S k are also the even moments of the standard Wigner matrix. 4For notational convenience, when discussing infinite (generalized) Wishart matrices we will simply refer to {W (c)} as the Wishart matrix even when the elements of Xn are i.i.d. but not Gaussian. We will occasionally add a subscript such as W1(c) to differentiate between different realizations of the ensemble {W (c)}. When discussing finite Wishart matrices, we will implicitly assume that the elements of Xn are i.i.d. Gaussian. Its use in either manner will be obvious from the context. 0