State the CLT, precisely

Question

State the central limit theorem precisely, including its assumptions, and name the most common ways it is misquoted.

Accepted Answer

Statement (classical i.i.d. version). Let X1, X2, \dots be i.i.d. random variables with mean \mu and finite variance \sigma^2 0. Then, as n 	o \infty, \frac{\bar{X}n - \mu}{\sigma/\sqrt{n}} \xrightarrow{d} \mathcal{N}(0, 1), i.e., the standardized sample mean converges in distribution to a standard normal. Equivalently, \sqrt{n}(\bar{X}n - \mu) \xrightarrow{d} \mathcal{N}(0, \sigma^2). Two assumptions do the work: (i) finite variance, and (ii) i.i.d. sampling, though independence can be relaxed (Lindeberg–Feller for non-identical terms, martingale and mixing CLTs for weak dependence). Common…

State the CLT, precisely

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