Proof of Central Limit Theorem

• chingkui
In summary, the Central Limit Theorem can be proven using the uniqueness of moment generating functions or the characteristic function. The characteristic function is a necessary step in existing proofs and it is a quick and convenient tool. However, it is not the only approach and Stein's method can also be used. For those interested in learning more, there are resources available such as the links provided.

chingkui

I have been reading some books about the proof of the Central Limit Theorem, all of them use the uniqueness of moment generating function. But since I have not yet seen a proof of the uniqueness properties, is there any proof that does not use this result? Thanks.

It can be done in terms of Characteristic functions also. A brief proof is given on the wikipedia site for Central Limit Theorem. Uniqueness of a characteristic function holds because it is just the Fourier transform of the corresponding density function, up to a multiplicative constant

chingkui said:
I have been reading some books about the proof of the Central Limit Theorem, all of them use the uniqueness of moment generating function. But since I have not yet seen a proof of the uniqueness properties, is there any proof that does not use this result? Thanks.

Moment generating functions are not unique in general. Proof of CLT uses characteristic function and CF's are unique.

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I am not familiar with the characteristic function, is there a place I can quickly read about its uniqueness? Is characteristic function a necessary step in existing CLT proofs? Thanks.

1) http://tt.lamf.uwindsor.ca/65-540/540Files/11.pdf [Broken]
2) http://tt.lamf.uwindsor.ca/65-540/540Files/13.pdf [Broken]

You need a lot of background to prove this result, which is why it's often skipped in undergraduate courses.

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ch.f is not the only tool for proving CLT, however in proper setting it is quick and convinient; as far as i know, stein's method another approach

What is the Central Limit Theorem?

The Central Limit Theorem states that when independent random variables are added, their sum tends toward a normal distribution (also known as a Gaussian distribution). This means that even if the individual variables are not normally distributed, the sum of those variables will be approximately normally distributed.

Why is the Central Limit Theorem important?

The Central Limit Theorem is important because it allows us to make inferences about a population based on a sample. This is because the sampling distribution of means will be approximately normal, regardless of the underlying distribution of the population. This makes it a powerful tool in statistics and data analysis.

What are the assumptions of the Central Limit Theorem?

There are three main assumptions of the Central Limit Theorem: 1) the random variables must be independent, 2) the sample size must be large enough (usually considered to be at least 30), and 3) the sample must be taken from a population with a finite standard deviation.

How is the Central Limit Theorem used in practice?

The Central Limit Theorem is used in many areas of data analysis, including hypothesis testing, confidence intervals, and regression analysis. It allows us to make inferences about a population based on a sample, and is used to determine the likelihood of certain outcomes or values.

Are there any limitations to the Central Limit Theorem?

While the Central Limit Theorem is a powerful tool, there are some limitations to its applicability. For example, it may not hold true for very small sample sizes or for non-independent data. Additionally, the Central Limit Theorem assumes that the population has a finite standard deviation, which may not always be the case.