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The central limit theorem may be established for the simple random walk on a crystal lattice (an infinite-fold abelian covering graph over a finite graph), and is used for design of crystal structures.
For the purposes of control limit calculation, the sample means are assumed to be normally distributed, an assumption justified by the Central Limit Theorem. The X-bar chart is always used in conjunction with a variation chart such as the x ¯ {\displaystyle {\bar {x}}} and R chart or x ¯ {\displaystyle {\bar {x}}} and s chart .
Comparison of probability density functions, () for the sum of fair 6-sided dice to show their convergence to a normal distribution with increasing , in accordance to the central limit theorem. In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve).
Then according to the central limit theorem, the distribution of Z n approaches the normal N(0, 1 / 3 ) distribution. This convergence is shown in the picture: as n grows larger, the shape of the probability density function gets closer and closer to the Gaussian curve.
Rayleigh fading is a reasonable model when there are many objects in the environment that scatter the radio signal before it arrives at the receiver. The central limit theorem holds that, if there is sufficiently much scatter, the channel impulse response will be well-modelled as a Gaussian process irrespective of the distribution of the individual components.
The law of iterated logarithms operates "in between" the law of large numbers and the central limit theorem.There are two versions of the law of large numbers — the weak and the strong — and they both state that the sums S n, scaled by n −1, converge to zero, respectively in probability and almost surely:
This section illustrates the central limit theorem via an example for which the computation can be done quickly by hand on paper, unlike the more computing-intensive example of the previous section. Sum of all permutations of length 1 selected from the set of integers 1, 2, 3
The characteristic function approach is particularly useful in analysis of linear combinations of independent random variables: a classical proof of the Central Limit Theorem uses characteristic functions and Lévy's continuity theorem. Another important application is to the theory of the decomposability of random variables.