Central limit theorem functional

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. 

Our next result concerns the central limit theorem, which places in evidence the remarkable behavior of the distribution function of when n is a large number. We shall now state and sketch the proof of a version of the central limit theorem that is pertinent to sums of identically distributed [p0i(x) = p01(a ), i — 1,2, ], statistically independent random variables. To simplify the statement of the theorem, we shall introduce the normalized sum s defined by... 

The central limit theorem thus states the remarkable fact that the distribution function of the normalized sum of identically distributed, statistically independent random variables approaches the gaussian distribution function as the number of summands approaches infinity—... 

The proof of the central limit theorem begins with the calculation of the characteristic function of... 

The multidimensional central limit theorem now states that the multidimensional characteristic function of sfn, -sj, behaves as follows ... 

Notice that those distribution functions that satisfy Eq. (4-179) still constitute a convex set, so that optimization of the E,R curve is still straightforward by numerical methods. It is to be observed that the choice of an F(x) satisfying a constraint such as Eq. (4-179) defines an ensemble of codes the individual codes in the ensemble will not necessarily satisfy the constraint. This is unimportant practically since each digit of each code word is chosen independently over the ensemble thus it is most unlikely that the average power of a code will differ drastically from the average power of the ensemble. It is possible to combine the central limit theorem and the techniques used in the last two paragraphs of Section 4.7 to show that a code exists for which each code word satisfies... 

If U0 and U1 were the functions of a sufficient number of identically distributed random variables, then AU would be Gaussian distributed, which is a consequence of the central limit theorem. In practice, the probability distribution Pq (AU) deviates somewhat from the ideal Gaussian case, but still has a Gaussian-like shape. The integrand in (2.12), which is obtained by multiplying this probability distribution by the Boltzmann factor exp (-[3AU), is shifted to the left, as shown in Fig. 2.1. This indicates that the value of the integral in (2.12) depends on the low-energy tail of the distribution - see Fig. 2.1. 

The physical and conceptual importance of the normal distribution rests on one unique property the sum of n random variables distributed with almost any arbitrary distribution tends to be distributed as a normal variable when n- oo (the Central Limit Theorem). Most processes that result from the addition of numerous elementary processes therefore can be adequately parameterized with normal random variables. On any sort of axis that extends from — oo to + oo, or when density on the negative side is negligible, most physical or chemical random variables can be represented to a good approximation by a normal density function. The normal distribution can be viewed a position distribution. 

If two Gaussian functions are convolved, the result is a gaussian with variance equal to the sum of the variances of the components. Even when two functions are not Gaussian, their convolution product will have variance equal to the sum of the variances of the component functions. Furthermore, the second moment of the convolution product is given by the sum of the second moments of the components. The horizontal displacement of the centroid is given by the sum of the component centroid displacements. Kendall and Stuart (1963) and Martin (1971) provide helpful additional discussions of the central-limit theorem and attendant considerations. 

The central-limit theorem (Section III.B) suggests that when a measurement is subject to many simultaneous error processes, the composite error is often additive and Gaussian distributed with zero mean. In this case, the least-squares criterion is an appropriate measure of goodness of fit. The least-squares criterion is even appropriate in many cases where the error is not Gaussian distributed (Kendall and Stuart, 1961). We may thus construct an objective function that can be minimized to obtain a best estimate. Suppose that our data i(x) represent the measurements of a spectral segment containing spectral-line components that are specified by the N parameters... 

From Eq. (A4) it follows that the limiting case a = 2 corresponds to the Gaussian normal distribution governed by the central limit theorem. For / = 0, the distribution is symmetric, y translates the distribution, and c is a scaling factor for X. Thus, y and c are not essential parameters if we disregard them, the characteristic function fulfills... 

There is an additional difficulty in experimental design where one or both of the C.I.E. Luminosity Functions are used as a reference function. The Central Limit Theorem insures that any results obtained through the addition to or subtraction from these reference functions will contain all of the errors associated with these functions. 

As we saw in Section 3.1.1, the familiar bell-shaped curve describes the sampling distributions of many experiments. Many distributions encountered in chemistry are approximately normal [3], Regardless of the form of the parent population, the central limit theorem tells us that sums and means of samples of random measurements drawn from a population tend to possess approximately bell-shaped distributions in repeated sampling. The functional form of the curve is described by Equation 3.19. 

The usual assumptions leading to the normal error probability function are those required for the validity of the central limit theorem The assumptions leading to this theorem are sufficient but not always altogether necessary the normal error probability function may arise at least in part from circumstances different from those associated with the theorem. The factors that in fact determine the distribution are seldom known in detail. Thus it is common practice to assume that the normal error probability function is applicable even in the absence of valid a priori reasons. For example, the normal error probability function appears to describe the 376 measurements of Fig. 3 quite well. However, a much larger number of measurements might make it apparent that the true probability function is slightly skewed or flat topped or double peaked (bimodal), etc. 

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]... 

The standard normal distribution function, with standard deviation equal to 1 and mean equal to zero, is presented in Figure 3.1. The confidence interval defined for an experimental set of data Xi,..., xn by fix ctx means that there is a 68.26 percent probability (see Section 3.1.4) that the correct value lies within the interval. There is a 95.44 percent probability that the correct value lies within the interval confidence interval fix 2ax- The Central-Limit Theorem described in Section 3.1.5 is often invoked to justify using the normal distribution as a basis for interpreting experimental data. 

An important issue is to verify that the energy differences are normally distributed. Recall that if the moments of the energy difference are bounded, the central limit theorem implies that given enough samples, the distribution of the mean value will be Gaussian. Careful attention to the trial function to ensure that the local energies are well behaved may be needed. 

This result is nothing other than the central limit theorem, since here r is the sum of independent random variables. We observe that the limiting law depends only on a single parameter (Nl2), When N - oo, the effects related to the microstructure of the links disappear and the result depends on the function p(u) only through the parameter l. Thus, in the limit N - oo, the behaviour of these chains with independent links, is universal, and the probability law... 

This result is known as the central limit theorem and serves to emphasize the importance and applicability of the normal distribution function in statistical data analysis since non-normal data can be normalized and can be subject to basic statistical analysis. ... 

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