# Papers or documents about the central limit theorem and its possible extensions: what happen when the sample size is big? [closed]

The central limit theorem in its most popular form states that (without being too formal) for a set of random variables $$X_1,X_2,...,X_n$$ independent and identically distributed with mean $$\mu$$ and standard deviation $$\sigma$$ we have that $$\sqrt{n}(\bar{X}-\mu)$$ converges in probability to a normal distribution with mean $$0$$ and standard deviation $$\sigma$$.

Then there are other kind of central limit theorem that apply under certain conditions to sequence of independent random variables with different means and standard deviations (Lindenberg).

What I can often read is that for statistical tests that relies on the fact that data are normally distributed, we do not have to care too much about the normality because thanks to the central limit theorem the results will remain valid. This is for example explained on Wikipedia for the one-sided Student's t-test if we utilize the law of large number additionally.

If I remember well, it is also the case for ANOVA.

Question Does it exist any documentations, website or papers about some statistical tests that suppose that data are normally distributed but works well when the sample size of observations is big ?

• For Q1 only (thus putting as comment) (The square root version of) Wald test? It starts from the same test statistic as a Student's $t$-test and relies on the fact that the test statistic is asymptotic normal thanks to CLT and Slutsky's. This question on "what is a big enough sample size" may also be helpful. Commented Jul 26, 2023 at 9:34
• "What I can often read is that for statistical tests that relies on the fact that data are normally distributed, we do not have to care too much about the normality because thanks to the central limit theorem the results will remain valid." ... consider an F test for equality of variances. It's a statistical tests that relies on the sample being from a normally distributed "population" but ... what's the argument that the CLT makes it "work" regardless of the population distribution you're sampling? Commented Jul 26, 2023 at 10:27
• Glen_n this is exactly why I am asking the question. I do not really see how and in what cases this apply. I know it works for t-test and now I also know it is possible for Wald test thanks to B.Liu. I hope to gather some resources with this topic about this subject. Commented Jul 26, 2023 at 11:42
• You have two questions here, should really ask them separately! Commented Jul 26, 2023 at 13:02
• I modified it, now there is one question. Commented Jul 26, 2023 at 13:16