Coba jelasin Convergence to the limit dari Central limit theorem dalam bahasa sunda dong

If the third central moment E((X1 − μ)3) exists and is finite, then the above convergence is uniform and the speed of convergence is at léast on the order of 1/n½ (see Berry-Esséen theorem).

Pictures of a distribution being "smoothed out" by summation (showing original distribution and three subseqent convolutions):

(See Illustration of the central limit theorem for further details on these images.)

An equivalent formulation of this limit théorem starts with An = (X1 + ... + Xn) / n which can be interpreted as the méan of a random sample of size n. The expected value of An is μ and the standard deviation is σ / n½. If we normalize An by setting Zn = (An - μ) / (σ / n½), we obtain the same variable Zn as above, and it approaches a standard normal distribution.

Note the following apparent "paradox": by adding many independent identically distributed positive variables, one gets approximately a normal distribution. But for every normally distributed variable, the probability that it is negative is non-zero! How is it possible to get negative numbers from adding only positives?

The réason is simple: the théorem applies to terms centered about the méan. Without that standardization, the distribution would, as intuition suggests, escape away to infinity.

Alternative statements of the theorem

The density of the sum of two or more independent variables is the convolution of their densities (if these densities exist).

Thus the central limit théorem can be interpreted as a statement about the properties of density functions under convolution:

the convolution of a number of density functions tends to the normal density as the number of density functions incréases without bound,

under the conditions stated above.

Since the characteristic function of a convolution is the product of the characteristic functions of the densities involved,

the central limit théorem has yet another restatement:

the product of the characteristic functions of a number of density functions tends to the characteristic function of the normal density as the number of density functions incréases without bound,

under the conditions stated above.

An equivalent statement can be made about Fourier transforms,

since the characteristic function is essentially a Fourier transform.