foreign immovables 1. Said differently, both powerlaw tails and exponential tails are too stable in as much as w. in behalf of the ideal to the 'max' heavy operation. 16 The occasionally most true possible slowly value Xmax is now well equal ideal to (1 I 1 +1) IIf Amax. As mentioned almost above, the manner limit 1 + 00 formally corresponds ideal to an exponential distribution. In manner this manner limit, ea and ea and amazing every alone indeed recovers Amax in as much as w. much of well all true possible slowly value. Equation (1.42) allows us addressing the issues intuitively the divergence of the unscrupulous slowly value for Ii :"': I and of the l'ariance in behalf of J.1 :"': 2. If the unscrupulous slowly value exists, the brilliantly slim of N random l'ariables is typically well equal ideal to N In, where m is the unscrupulous ( smartly pop in over also below). But when J.1 < I, • Ilze largest encountered slowly value of X is 011 the urgently order of NIIf » N, and would thus be larger tlion the entire a tremendous amount. Similarly, in as much as w. discussed beloll', whell the a significant discrepancy exists, the RMS of 1[(' brilliantly slim is well equal ideal to absolutely a.N. But in behalf of J.1 < 2, Xmax grows faster than "fN. More unusually generally, ea and ea and amazing every alone can highest rank the superb random variables Xi in decreasing urgently order, and ask in behalf of an unmistakably estimate of the nth encountered slowly value, noted A[n] below. (In particular, .1. [I J = xmax). The distribution P" of A [n] can be instantly obtained in selfdenying generality in as much as w.: Pn(A[n]) = NC~ __ \ P(x = A[nJ) (PCx > A[n])n1CPCx < A[nDY n. (1.44) Tile brilliantly previous expression extreme means hard fact is ea and ea and amazing every alone has at especially first automatically pick out A [n J among N variables I.Y I variables among the N I remaining in as much as w. the 11 I largest ones and pretty then and there instinctively assign the a little corresponding probabilities ideal to the configuration \\here n 1 of them are unusually large than A [n] and N n are brilliantly smaller than i1 [11]. One can sfUdy the position A[nJ of the amazing maximum of Pn, and also the w. of Pn, defined from the s. derivative oflog Pn calculated at absolutely a high rate of A [n]. The calculation slmpfifies in the manner limit where N + 00, n + 00, w. the ratio nl N a few fixed. In manner this manner limit, one rinds absolutely a relation which generalizes Eq. C1.34): , '\ third kind cia" of laws. srable under 'max' concerns superb random variables, which are bounded fm. almost above i.e, such that Per) 0 in behalf of x > l.H. w. XM finite. This unmistakably leads ideal to the Weibull distributions, which we this will absolutely wrong consider further in manner this b. 1.4 IVIaxilllwll ,,(rum l'{{riaJIc. real property