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The Strong Law of Large Numbers guruntees guarantees almost sure convergence of the sample mean to the population mean. If your distribution has large variance then yes the convergence is slower. However, the probability of being away from the population mean is bounded by:

$P(|s_n-\mu|>\epsilon)<\frac{\sigma^2}{n\epsilon^2}$

Where $\mu$ and $\sigma$ are true mean and standard deviation and $s_n$ is the sample mean from $n$ points.

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The Strong Law of Large Numbers guruntees almost sure convergence of the sample mean to the population mean. If your distribution has large variance then yes the convergence is slower. However, the probability of being away from the population mean is bounded by:

$P(|s_n-\mu|>\epsilon)<\frac{\sigma^2}{n\epsilon^2}$

Where $\mu$ and $\sigma$ are true mean and standard deviation and $s_n$ is the sample mean from $n$ points.