Therefore $ hat{ sigma}^2$ is biased for any finite sample size. We can also easily derive that $${ rm var}( hat{ sigma}^2) = frac{ 2 sigma^4(n 1)}{n^2}$$ From these facts we can informally see that the distribution of $ hat{ sigma}^2$ is becoming more and more concentrated at $ sigma^2$ as the sample size increases since the mean is converging to $ sigma^2$ and the variance is converging to $0$. (Note: This does constitute a proof of consistency, using the same argument as the one used in the answer here).

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