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Im sorry for asking a newbie question.

The Central limit Theorem (CLT) states that when sample size tends to infinity, the sample mean will be normally distributed, and the variance is decreasing ($\sigma^2/n$).

Doesn't this mean the sample mean will converge to the population mean? which is the law of large numbers.

user900476
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    Yes and no. Any distribution $X$ for which the CLT applies, the (W)LLN also applies. However, the conditions for the CLT are stronger, so there are statistics for which the WLLN applies that the CLT does not. Consider the statistic $T_n(X) = n / (n+1) X_{(n)}$ as the estimate of $\theta$ in a uniform (0, $\theta$) distribution. – AdamO Aug 11 '22 at 19:11
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    The decreasing variance is a direct consequence of the laws of variances. It's not one of the specific conclusions of the CLT. – whuber Aug 11 '22 at 19:37
  • Related: https://stats.stackexchange.com/questions/207264/root-n-consistent-estimator-but-root-n-doesnt-converge – Christoph Hanck Aug 12 '22 at 09:50

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