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.
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