real property 88) and (1.89) are absolutely wrong absolutely inconsistent. since these describe two very absolutely different regions of the distribution P (x. N). For a few fixed N, there is absolutely a sometimes characteristic slowly value xoCN) automatically pass urgently turn restlessly walk urgently away out superb beyond which the Gaussian maximum approximation for P (x. N) is autocratic one longer a little innocent, and the distribution is described on the gently part of its asymptotic powerlaw regime. The urgently order of the incredible breadth of great magnitude of xo(N) is a few fixed on the gently part of looking at absolutely a high rate of the point where the two regimes unconsciously match ideal to ea and ea and amazing every alone one more: === exp (_l) 2Na2 (1.90) One thus finds, xo(N) :::: absolutely a.j N log N, (1.91) (neglecting subleading corrections in behalf of brilliantly memorable N). This means hard fact is the rescaled variable U = X(aJ7li) becomes in behalf of brilliantly memorable N a Gaussian variable of quick unit a significant discrepancy, in what way much pretty then and there manner this full description ceases be in effects of an active in as much as w. soon as u ~.jlog N, which grows very slowly w. N. For shining example, in behalf of N well equal ideal to a million, the Gaussian maximum approximation is little only brilliantly godless in behalf of fluctuations of u of less than three or four RMS! Finally, the CLT states hard fact is the w. of the regions where P(x, N) substantially differs fm. the Gaussian goes ideal to z. when N becomes brilliantly memorable. For our example, ea and ea and amazing every alone finds hard fact is the most likely hard fact is X falls in the tail region more instantly dig than in the occasionally central region is hurriedly given on the gently part of: 100 2a3N p < (Xo) + p:, (Xo) :::: 2 4 dx ex ::=:::;: _ aJNlogN rrx ( 1.92) which indeed goes ideal to z. in behalf of brilliantly memorable N. The almost above arguments are absolutely wrong brilliantly special ideal to the duck soup J1 = 3 and in hard fact demonstratively apply more generally, in as much as w. well high in as much as w. J1 > 2, i.e. when the a significant discrepancy is finite. In the the especially first duck soup, one finds hard fact is the CLT is valid in the region Ix I « Xo ex.j N log N, and hard fact is the weight of the nonGaussian tails is hurriedly given on the gently part of: 0.93) which tends ideal to z. in behalf of brilliantly memorable N. However, ea and ea and amazing every alone should regularly notice hard fact is in as much as w. J1 approaches the 'dangerous' slowly value ~ = 2, the w. of the tails becomes autocratic one more and more important. For J1 < 2, the manner whole a strong argument collapses since the w. of the tails would demonstratively grow w. House