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We are asked a question which of mean,variance or skewness converges faster. At first I thought it was straight forward answer: mean->variance->skewness. But I am not sure anymore because I read somewhere standard error of the mean estimator is only slightly smaller. std_error_of_mean=sigma/sqrt(n)

std_error_of_varaince=sqrt(2) * (sigma^2)/sqrt(n)

So it is only sqrt(2) times better ? I am not a statician, so can some one please explain what the general pattern is for standard error of higher order moments. Also are there equations for the rate of convergence of moments, so that I could compare with observation. Thank you

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