All,
I'm wondering if there are any notable, basic discrete probability distributions with "fat/heavy tails" or a large kurtosis? I know the Geometric Distribution's excess kurtosis approaches 6, but I can't find any that are larger.
Are there any discrete distributions where the kurtosis ( or other higher normalized moment ) is infinite / undefined?
( By basic, I guess I mean something one could easily implement with game tokens like dice or cards. )
For a "fat tail" you might take probability mass function $p(x) = \zeta(s)^{-1}/x^s$ for positive integers $x$, where $s > 1$. This has finite variance for $s > 3$, and finite kurtosis for $s > 5$. As $s \to 5+$ the excess kurtosis is, according to Maple, $ \dfrac{8100 \zeta(5)}{8100 \zeta(3) \zeta(5) - \pi^8} (s - 5)^{-1} + O(1)$, where $8100 \zeta(5)/(8100 \zeta(3) \zeta(5) - \pi^8) \approx 13.82154183$.
For something you can implement easily with a fair coin, you might try this. Let $N$ be the number of flips of the coin until the first heads, and $X = r^N$. For the kurtosis to be finite, you need $r < 2^{1/4}$. According to Maple, the excess kurtosis is $${\frac {-13\,{r}^{7}+16\,{r}^{5}+2\,{r}^{4}+8\,{r}^{3}+16\,{r}^{2 }-16}{2 \left(2- {r}^{4}\right) \left(2- {r}^{3} \right) }}$$ For example, if $r=1.1$ the excess kurtosis is approximately 24.22623314.