What distribution results in adding two Pareto distributions

Question

I'm wondering what distribution results in adding two (or more) type-one Pareto distributions of the form $x^{-\alpha}$. Experimentally, it looks like a two-mode power-law, asymptotic to the difference of alphas.

Answer 1

Edited to be a bit more readable. Distributions add by convolution. The Pareto distribution is piece wise defined as a $k^a x^{-a-1}$ for $x\geq k$ and 0 for $x<k$. The convolution of two Pareto functions $k^a x^{-a-1}$ and $j^b x^{-b-1}$ is:

$$ a (-1)^{-b} b k^a j^b \Gamma (a+b+1) \times \\ \left(\left(\frac{1}{t-j}\right)^{a+b+1} \, _2\tilde{F}_1\left(b+1,a+b+1;a+b+2;\frac{t}{t-j}\right)- \\ \left(\frac{1}{k}\right)^{a+b+1} \, _2\tilde{F}_1\left(b+1,a+b+1;a+b+2;\frac{t}{k}\right)\right), $$

where $j+k<x$ and 0 for $x\leq j+k$, which although complex field within the that term, is real valued outside of it. $\, _2\tilde{F}_1(w,x;y;z)$ is Hypergeometric2F1Regularized here in Mathematica code. Not all choices for the parameters will yield positive valued density functions. Here is an example of when they are positive. For the two Pareto distributions let a = 2, b = 3, j = 0.1 and k = 0.3.

and their plots are in blue for the {k, a} function and in orange for the {j, b} function. Their convolution is then graphically

which, when the tails are examined looks like

where the green is the convolution.

From your question, you may be asking about the ordinary addition of two Pareto distributions. In that case, the area under the curve is two, so the sum is not a density function, which needs to have an area under the curve of one. However, if that is the question then $\frac{a k^a t^{-a-1}+b j^b t^{-b-1}}{t^{a-b-1}}$ for $b>a>0$ simplifies to $t^{-2 a} \left(b t^a j^b+a k^a t^b\right)$, which has a limit of $a k^a$ only if $b=2a$, and is 0 or infinity in all other cases. In other words, the arithmetic sum of two Pareto distributions only has tails that are the difference between $a$ and $b$ when $b=2a$, and the arithmetic sum is not a density function, and the sum would have to be scaled for two probabilities, $1=p+q$ in order to be a density function. Although arithmetic addition of density functions to define another density function does occur, it is unusual. One example of this occurs in pharmacokinetics, where the sum of two or more exponential distributions is used to define a density function. To make a long story short, that is not something I would recommend.

Hopefully this answers your question. If it does not, please object to my answer or please add some more information.