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Let $X_1$, $X_2$, ... be iid random variables with $E(X_1)$ = $\mu < \infty$ and $Var(X_1)$ = $\sigma^2 < \infty$.

Does the Central Limit Theorem implies that the sample mean converges in distribution to a normally distributed random variable?

Also, does the theorem implies that the sample mean is a consistent estimator for $\mu$?

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  • $\begingroup$ Is there any difference between your question 1 and your question 2? $\endgroup$ – Watson Feb 14 '16 at 14:58
  • $\begingroup$ Oppss.. I've amended my question! $\endgroup$ – OinkOink Feb 14 '16 at 15:01
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    $\begingroup$ Sample mean converges to a degenerate $N(\mu,0)$ - which is LLN and implies consistency of the estimator. CLT states that properly scaled sample mean converges to $N(\mu,\sigma^2)$ and you can derive (weak) LLN from CLT. $\endgroup$ – A.S. Feb 20 '16 at 8:06
  • $\begingroup$ @A.S. Actually, the CLT establishes the convergence to a centered normal random variable, not to $N(\mu,\sigma^2)$ (except if $\mu=0$). $\endgroup$ – Did Jul 29 '16 at 21:21

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