Density distribution Gaussian limit

One of the fundamental results in probability is the Central Limit Theorem which concerns an infinite collection of mutually independent, identically distributed random variables. The basic observation is that if we have some random variables drawn from some particular distribution we may define a new random variable by averaging. The average will not have the same distribution as those of the collection, but it will have a well defined density. The Central Limit Theorem effectively describes the limiting density obtained by averaging over an infinite collection. Remarkably, independent of the density of a particular member of the collection, this averaged random variable will ultimately be Gaussian with mean and variance specified by the mean and variance associated to the members of the collection. 

For chains having fewer than 50 bonds, such as the short chains in a network having bimodal distribution of lengths, for example, the distribution departs markedly from the Gaussian limit. A distribution that has been employed for short chains is the Fixman-Alben density distribution given by... 

Assuming Gaussian density distributions, we obtain in the limit of a complete overlap, rc,i = Tc,2, an expression identical to the internal excluded volume interaction energy as given by Eq. (2.91), apart from a factor 1/2. Omitting the numerical prefactor of order unity we can write... 

Remark. The white noise limit is not sufficiently defined by just saying rc 0. We have to construct a sequence of processes which in this limit reduce to Gaussian white noise. For that purpose take a long time interval (0, T) and a Poisson distribution of time points Ta in it with density v. To each Ta attach a random number ca they are independent and identically distributed, with zero mean. Consider the process... 

In the continuum limit the second moment increases linearly with time and in direct proportion to the diffusion coefficient, so that the probability density becomes the familiar Gaussian distribution for Einstein diffusion ... 

It is evident from the inverse power-law form of the probability density given by Eq. (93) that the second moment (x2) of the Levy a-stable distribution diverges since a + 1 <3. Equally clear is the fact that the first moment for this distribution diverges for a + 1 <2. The first and second moments converge for ot = 2, in which case the Levy stable distribution becomes a Gaussian distribution and the central limit theorem again applies to the time series. 

The tails prevent [19] convergence to the Gaussian distribution for N -= oo, but not the existence of a limiting distribution. These distributions as we have seen are called stable distributions. If the concept of a Levy distribution is applied to an assembly of temporal random variables such as the x, of the present chapter, then w(x) is a long-tailed probability density function with long-time asymptotic behavior [7,37],... 

It is important to spell out the limitations on the derivation of the distribution (2.2S) of fluctuations. Consider the most general initial state, which is XPinitial state is pure if the rank of the matrix p is unity. Otherwise, it is a mixture. The transition intensity to the final state / is y = Y,ijx Pijxj where x = < T />. y = x px is then a quadratic form where the amplitudes x have a gaussian probability density... 

This figure shows a plot of the scaled probability density of a hydro-genic ion for many dimensions ranging from D = 2 to D = 100. One can readily see that as D increases, the electron-nucleus cusp in the wavefunction in the region close to iZ = 0 is very much de-emphasised. But at the same time, as D increases the probability distribution becomes more and more sharply peahed about its most probable value. In the limit as D oo, the probability distribution approaches a Gaussian corresponding to the zero-point vibrational motion in the effective potential. 

This fact says that p x, t, x, t ) is a special probability density whieh evolve with the increasing of tome (at from the local Dirac distribution (at When an initial Gaussian distribution is considered (sinee all continuous distributions can be eventually reduced to a Gaussian form, according with the central limit theorem)... 

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