# Understanding Central Limit Theorem vs. Law of Large Numbers

I am trying to clarify these two concepts - and understand the differences between the Central Limit Theorem (https://en.wikipedia.org/wiki/Central_limit_theorem) and the Weak Law of Large Numbers (https://en.wikipedia.org/wiki/Law_of_large_numbers).

As an example, suppose I have a coin and I don't know the true probability of Heads or Tails - I start to flip the coin again and again:

• The Law of Large Numbers states that if I flip this coin enough times, I will get an estimate of the true probability of getting a Heads
• The Central Limit Theorem states that as I flip the coin again and again, the distribution for the probability of getting a Heads will follow a Standard Normal Distribution

Is my understanding of this correct?

Thanks!

• No, your interpretation of the WLLN is not correct, but it is a correct interpretation of SLLN, although it should really read a very good estimate, since any function of the data is an estimate. As for CLT, what does the distribution for the probability of getting heads mean? The probability of getting heads is a fixed number. Dec 22, 2022 at 22:00
• Crucial thing in CLT is normalization. This theorem cannot be described by a loose language like the one you have used. Dec 22, 2022 at 23:20

The Linderberg-Levy CLT, teaches us that for an iid sample of a variable with finite expected value and variance, $$\sqrt{n}(\bar{X}_n - \mu)\rightarrow_d N(0,\sigma^2)$$ where $$\bar{X}_n=\dfrac{1}{n}\sum x_i$$. Notice that the value of $$\bar{X}_n$$ depends on $$n$$, i.e. the sample size. In other words, the random variable converges in distribution to the normal, and this is true independently of the initial distribution.