Central limit theorem generalized

Systems that display strange kinetics no longer fall into the basin of attraction of the central limit theorem, as can be anticipated from the anomalous form (1) of the mean squared displacement. Instead, they are connected with the Levy-Gnedenko generalized central limit theorem, and consequently with Levy distributions [43], The latter feature asymptotic power-law behaviors, and thus the asymptotic power-law form of the waiting time pdf, w(r) AaT /r1+a, may belong to the family of completely asymmetric or one-sided Levy distributions L+, that is,... 

The justification for the use of the lognormal is the modified Central Limit Theorem (Section 2.5.2.5). However, if the lognormal distribution is used for estimating the very low failure frequencies associated with the tails of the distribution, this approach is conservative because the low-frequency tails of the lognormal distribution generally extend farther from the median than the actual structural resistance or response data can extend. 

When we are dealing with samples rather than populations, we cannot use the standard normal deviate, Z, to make predictions since this requires knowledge of the population mean and variance or standard deviation. In general, we do not know the value of these parameters. However, provided the sample is a random one, its mean 5 is a reliable estimate of the population mean p, and we can use the central limit theorem to provide an estimate of o. This esti mate, known as the standard error of the mean, is given by ... 

One can also view (29) as a generalization of Cramer s LDT [62]. Recall that this theory treats the wings of a distribution like V(m) correctly, to which the central limit theorem (CLT) does not apply [63]. However, the CLT approximation in this problem has quite a beautiful interpretation and is worth describing separately. It can be derived by assuming that n (cr) is itself a Gaussian in Eq. (29), or just written down directly ... 

The result decreases with k less quickly than S contrary to the unperturbed and the collapsed chain [see Eqs. (2.1.73) and (2.2.14), respectively] as a consequence, the sum of the long-range bond correlation contributions diverges and the mean-square chain size increases more quickly than N -. In particular, the central-limit theorem cannot be applied and the Gaussian distribution for W(k, r) is incorrect. Actually, it has been determined that for /c > 1 this function obeys the generalized Domb-Gillis-Wilmers form [71, 72]... 

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