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I am confused about two assumptions of Central Limit Theorem that are as follows. Could you please clarify why do we need these two assumptions for CLT to be valid. And what would it be if we fail to meet one of the assumptions. Thanks so much.

Independence Assumption: The sample values must be independent of each other.

10% Condition: When the sample is drawn without replacement (usually the case), the sample size, n, should be no more than 10% of the population.

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    $\begingroup$ Does $n \to \infty$ sound like 10% ? $\endgroup$ – Tim Mar 12 '18 at 9:32
  • $\begingroup$ An extreme example of non-independence is illustrated by the lazy sampler. This individual observes the random variable $X_1$ but--instead of observing the sequence $X_2, X_3, \ldots,$--simply copies down the value of $X_1$ again and again and again. Such a sequence is random, because the first value is random. However, you can't even get started applying the CLT because the standard deviation of the values always is zero. If you ignore the standardization, the mean of the values is constantly $X_1$, which converges to $X_1$ rather than the mean of $X_1$'s distribution. $\endgroup$ – whuber Mar 12 '18 at 14:45
  • $\begingroup$ @Even It's a little unclear what you're asking. The relevant wikipedia page mentions several different central limit theorems. It doesn't seem you're actually asking about what any of them says. Can you give some context for your question? What in fact have you been told the CLT says and where is this written? I expect what you're actually discussing is something else (which may arguably be somewhat related to a CLT but is not actually about any of the forms of central limit theorem). $\endgroup$ – Glen_b -Reinstate Monica Mar 13 '18 at 4:54
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Central limit theorem's (there are several) concern with a sequence of $n$ independent and identically distributed random variables (so, yes, they need to be independent) and describe what happens when $n \to \infty$. It says nothing about size of sample drawn from the population and certainly nothing about sample size being too big. It is a theorem, so the assumptions need to be met, if they don't, you cannot assume that it applies.

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