Central limit theorem, derivation

After considerable algebraic manipulations, similar to the central limit theorem derivation in Chapter 2, the result is the Gaussian distribution inEq. 11.10. 

The Characteristic Function.—The calculation of moments is often quite tedious because of difficulties that may be encountered in evaluating the pertinent integrals or sums. This problem can be simplified quite often by calculation of the so-called characteristic function of the distribution from which, as we shall see, all moments can be derived by means of differentiation. This relationship between the characteristic function and moments is sufficient reason for studying it at this time however, the real significance of the characteristic function will not become apparent until we discuss the central limit theorem in a later section. 

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 ... 

One conclusion derived from the central limit theorem is that for large samples the sample mean X is normally distributed about the population mean p with var-iance ax, even if the population is not normally distributed. This means that we can almost always presume that X is normally distributed when we are trying to estimate or make a test on p, providing we have a large sample. 

A published derivation of the Green-Kubo or fluctuation-dissipation expressions from the combination of the FR and the central limit theorem (CLT) was finally presented in 2005. This issue had been addressed previously and the main arguments presented, but subtleties in taking limits in time and field that lead to breakdown of linear response theory at large fields, despite the fact that both the FR and CLT apply, " were not fully resolved. ... 

Williams and Evans have tested the transient steady state ES FR on glassy systems. They verified that while the transient ES FR is obeyed at all times, the ES FR converges slowly, and the convergence time increases as the glass transition point is reached. Since the steady state ES FR in conjunction with the central limit theorem can be used to derive the linear response expressions for transport properties, this indicates that the strength of the field for which linear response theory applies becomes smaller as this point is approached. The slow convergence of the ES FR is in accord with its derivation which relies on decay of correlation times for convergence. 

Thus, as given by Eq. (1.42), the probability distribution function for the end-to-end vector R is Gaussian. The distribution has the unrealistic feature that R can be greater than the maximum extended length Nb of the chain. Although Eq. (1.42) is derived on the freely jointed chain model, it is actually valid for a long chain, where the central limit theorem is applicable, except for the highly extended states. 

A more rigorous definition of uncertainty (Type A) relies on the statistical notion of confidence intervals and the Central Limit Theorem. The confidence interval is based on the calculation of the standard error of the mean, Sx, which is derived from a random sample of the population. The entire population has a mean /x and a variance a. A sample with a random distribution has a sample mean and a sample standard deviation of x and s, respectively. The Central Limit Theorem holds that the standard error of the mean equals the sample standard deviation divided by the square root of the number of samples ... 

Although the above derivation is for the freely jointed chain, the result actually bolds more generally. In general it can be shown that provided the conformational distribution is described by eqn (2.15), the distribu tion of the end-to-end vector H of a long diain (Af 1) is given by eqn (2.26). This is a result of the central limit theorem in statistics. ... 

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