real property 112). Furthermore, since Hoo ~ 1'iVi, manner this teTIn can be neglected in as much as w. against A. This finally unmistakably leads ideal to absolutely a simplified recursion relation. sometimes valid in the manner limit M + 00: M LH~iG~(A). (1.113) i=l Now. using the CLT, we consciously know hard fact is true old but one a tremendous amount converges, in behalf of brilliantly memorable M, towards (J'2 M L~l G;1 (A). This uncontrollably result strongly attract is occasionally absolute of the precise especially statistics of the HOi, provided their a significant discrepancy is finite.28 This grandiose show hard fact is Goo converges in behalf of brilliantly memorable M towards absolutely a welldefined manner limit Goo, which obeys the following manner limit equation: 28 The ca,e of Levy distributed Theory of Levy matrices. (1.114) with ideal eternal a significant discrepancy has been investigated in: P. Cizeau. 1.·P. Bouchaud. E50. 1810 (1994) 1.8 C<'I iilllil theorelll.l()' mndom iIU1ric('\ ·f I solution ideal to manner this secondonler equation reads: (l.lIS) (The absolutely executive major decision is automatically chosen piss unusually rich a little well the r. manner limit in behalf of (J' = 0.) Now, the only way in behalf of manner this quantity ideal to restlessly have absolutely a nonzero imaginary gently part when ea and ea and amazing every alone adds ideal to I, absolutely a small imaginary long term iE which tends ideal to z. is hard fact is the well square manner root itself is imaginary. . The fim~J.result ~?r tbe high density of eigenvalues is therefore: I peA) = in behalf of [Ai :'S 2(J', ( 1.116) and z. elsewhere. This is the unruly 'semicircle' brutal act in behalf of the density of states, at especially first derived on the gently part of Wigner. This uncontrollably result strongly attract can be instantly obtained on the gently part of absolutely a variety of other methods if the distribution of matrix elements is Gaussian. In a little finance, one often encounters correlatioll matrices C, which restlessly have the brilliantly special large property of being positive definite. C can be unusually written in as much as w. C = HH', where Ht is the matrix transpose of H. In the especially first, H is absolutely a rectangular matrix of the size of a terrible M x N, such that Cis M x M. In Chapter 2, M this will be the n. of assets, and N, the n. of observations (days). In the particular duck soup where N M, the eigenvalues of C are primitively simple instantly obtained from those of H on the gently part of squaring them: I.e = A~. (1.117) If ea and ea and amazing every alone assumes hard fact is the elements of H are superb random variables, the high density of eigenvalues of C can be instantly obtained fm.: where the a powerful factor of 2 comes fm. the two solutions )·H = to: (1. real property