# Relationship between kurtosis and the center of a distribution

Some have stated that kurtosis is the "movement of probability mass from the “shoulders” of a distribution into its center and tails" where "center" is defined as the range between $\mu \pm\sigma$. I was trying to find a proof for this but was unsuccessful. Can anyone point me to a proof of whether (i) larger kurtosis implies greater probability in the $\mu \pm\sigma$ range, or (ii) greater probability in the $\mu \pm\sigma$ range implies larger kurtosis?

• Given that you already know the answer to this, I think a more broadly beneficial question would perhaps ask whether anything can be said about more in the tails or more near the peak when kurtosis increases. – Glen_b -Reinstate Monica Nov 25 '17 at 9:48
• If my internet connection stays up, I should have time to get back to this over the next couple of days. Which definition of more in the tails is that? – Glen_b -Reinstate Monica Dec 7 '17 at 1:14
• The 3 theorems are (i) kurtosis is between E(Z*4*I(|Z|>1) and E(Z^4*I(|Z|>1) +1, for all distributions with finite 4th moment, (ii) when the distributions are restricted to continuous with decreasing density of Z^2 on (0,1), then the +1 can be replaced with +.5, and (iii) for any sequence of distributions having kurtosis tending to infinity, E(Z^4*I(|Z|>b))/kurtosis -> 1, for all real b. The definitions of tailweight implicit in these 3 theorems are, for (i) and (ii), E(Z*4*I(|Z|>1), and for (iii), E(Z*4*I(|Z|>b). – Peter Westfall Dec 12 '17 at 1:14

I will try to dig up the reference, but I don't think you can prove what you claim. Imagine taking a normal distribution, then take all points within 1-$\sigma$ and put them out at >3-$\sigma$. That will have more kurtosis, I believe, but zero mass in the center.