# discrete pareto distribution?

I always thought, that Pareto distriubtion is continuous.

I found a paper, that states that $$P(X=c)=\frac{1}{c^{\alpha}} - \frac{1}{{c+1}^{\alpha}}$$

for $$c=1,2,3,...$$, where $$X$$ is a Pareto random variable.

Is that right?

Maybe they meant an other distribution?

This seems like a natural discretization of the continuous Pareto distribution. I agree it's not what people typically mean. $$P(X=c)$$ in that paper is simply the probability under the true Pareto distribution that $$c\le X\le c+1$$. The key characteristic of the Pareto distribution in many cases is that $$P(X>c)=O(1/c^\alpha)$$, which holds here.
• Thank you very much. Is there any way to compute the convolution of n $X_i$, which are i.i.d,meaning: $P(X_1+...+X_n=j)$. Can I make any statement on the common distribution?
• I don't think there's a closed form for the sum of Paretos. However, the Pareto distribution is subexponential, so you'll get that $P(X_1+X_2>j)\sim 2P(X_1>j)$, i.e. the ratio of those two objects is 1 as $j \rightarrow \infty$. Oct 29, 2020 at 17:20