# Convex combination of Dirichlet random variables

For positive integer $$k$$, let $$(X_1,\ldots,X_k)\sim\mathrm{Dir}(\alpha_1,\ldots,\alpha_k)$$ be a probability distribution over $$k$$ items drawn from a $$k$$-component Dirichlet distribution and $$p=(p_1,\ldots,p_k)$$ be another fixed distribution. What is the pdf of the random variable $$Q=\sum_{i=1}^k p_i X_i$$?

If each $$X_i$$ were independent gamma random variables with parameter $$\alpha_i$$, i.e., $$X_i\sim\mathrm{Gamma}(\alpha_i,\beta)$$, then this would be easy: by linearity, $$Q\sim\mathrm{Gamma}(\sum_{i=1}^k p_i\alpha_i, \beta)$$, following the notations in this lecture note. The Dirichlet random variables can be obtained by normalizing the gamma random variables, and the marginal of each component is a beta random variable. A way to show this is the case is done by observing that if $$Y_i\sim\mathrm{Gamma}(\alpha_i,1)$$ for $$i\in\{1,2\}$$, then $$\frac{Y_1}{Y_1+Y_2}\sim\mathrm{Beta}(\alpha_1,\alpha_2)$$. This requires that $$Y_1$$ and $$Y_2$$ are independent from this post.

I suspect that $$Q$$ is a beta random variable, since it looks like a convex combination of gamma random variables up to normalization. An obstacle that prevents me from showing this is that for $$X_i\sim\mathrm{Gamma}(\alpha_i,\beta)$$, $$\sum_{i=1}^k p_i X_i$$ is no longer independent of $$X_1+\ldots+X_k$$ (unless $$p$$ has some special form). Was convex combination of Dirichlet components studied before? Any comments will be appreciated.

There is no well-known distribution for the weighted sum of a random vector with a Dirichlet distribution. However, as partially checked in this old answer, the beta distribution can be a good approximation for it (you do not need to normalize the vector $$p$$ as it is already normalized).

This 2023 paper derives a novel integral representation for the density of a weighted sum of Dirichlet distributed random variables (Appendix A.1, page 15); you can use it if you want the exact distribution. This paper also presents various non-asymptotic Gaussian-based bounds for probabilities of linear transformations of a Dirichlet random vector.

Regarding your results: Note that for $$c>0$$, and $$X$$ and $$Y$$ that are independent with

$$X \sim \text{Gamma} (\alpha_1, \lambda), Y \sim \text{Gamma} (\alpha_2, \lambda),$$

we have

$$cX \sim \text{Gamma} \left ( \alpha_1, \frac{\lambda_1}{c} \right )$$

$$X +Y \sim \text{Gamma} \left (\alpha_1+\alpha_2, \lambda \right).$$

Hence, generally there is no $$\alpha'$$ and $$\lambda'$$ such that $$p_1X+p_2Y \sim \text{Gamma} \left (\alpha', \lambda' \right),$$ unless $$p_1=p_2=p$$, for which we have $$pX+pY \sim \text{Gamma} \left (\alpha_1+\alpha_2, \frac{\lambda}{p} \right).$$