LLN versus CLT

Problem

For i.i.d. variables with mean $\mu$ and finite variance $\sigma^2$ , state what the law of large numbers says about the sample mean $\overline{X}_n$ and what the central limit theorem says about its standardized error.

Reveal answer or reference solution

The law of large numbers says

\overline{X}_n \xrightarrow{p} \mu.

The central limit theorem says

\frac{\sqrt{n}(\overline{X}_n-\mu)}{\sigma} \xrightarrow{d} \mathcal{N}(0,1).

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Probability Practice