Kurtosis is seen as a measure of the discrepancy between the observed data and a Gaussian distribution  . In this work an empirical study is conducted to investigate the behaviour of the sample estimate of kurtosis with respect to sample size and the tail index  . The study will be focus on samples from the symmetric stable distributions. It was found that the expected value of excess kurtosis divided by the sample size is finite for any value of the tail index and the sample estimate of kurtosis increase as a linear function of sample size and tail index  .
Kurtosis is seen as a measure of the discrepancy between the observed data and a Gaussian distribution and is defined when the 4th moment is finite . In this work an empirical study is conducted to investigate the behaviour of the sample estimate of kurtosis with respect to sample size and the tail index when applied to heavy-tailed data where the 4th moment does not exist . The study will focus on samples from the symmetric stable distributions. It was found that the expected value of excess kurtosis divided by the sample size is finite for any value of the tail index and the sample estimate of kurtosis increases as a linear function of sample size and tail index . It is very sensitive to changes in the tail-index .
A
and is defined when the 4th moment is finite
A
when applied to heavy-tailed data where the 4th moment does not exist
D
be
R
increase
increases
A
. It is very sensitive to changes in the tail-index