Can you show that $\bar{X}$ is a $\sqrt{n}$-consistent and strongly consistent for $\mu$? Where $X_1, X_2,..., X_n$ be iid from $P\in{\wp}$ and $\mu$, mean of $P$ is assumed to be finite.
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What is the mean of $\mathbf{P}$? May be you wanted to say the mean of $X_{i}$? – ABK Aug 26 '20 at 13:52
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related: https://stats.stackexchange.com/questions/207264/root-n-consistent-estimator-but-root-n-doesnt-converge/207281#207281 – Christoph Hanck Aug 26 '20 at 14:14
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The statements are the weak and strong laws of large numbers, respectively.
The proofs can be found, for example, in
https://www.math.ucdavis.edu/~tracy/courses/math135A/UsefullCourseMaterial/lawLargeNo.pdf
ABK
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