Distribution without Kurtosis I am doing work on predicting stock price volatility with ARCH Models. One thing struck me. It was mentioned that the Model only had a Kurtosis under certain restrictions. How is it possible for a distribution not to have a Kurtosis, isn't that counterintuitive? The report stated that the Kurtosis only existed under the condition that: $1-\kappa_z a_1 > 0$ I will not bother with the details of the parameters since I believe it to be irrelevant. I simply wonder how that is possible and what it concludes.

Let $c = \sum_{n=1}^\infty \frac{1}{n^5} < \infty$ and let $X$ be a discrete random variable with $P(X=n) = \frac{1}{c n^5}$ for integers $n \ge 1$. Then $E[X^4] = \infty$ so the kurtosis of $X$ is undefined.

The related Cauchy distribution is a classic example that doesn't even have an expectation, let alone kurtosis.