Is there a way to approximate the convolution of Beta variables?

Specifically, I am trying to find an approximation to $g(x_0)$:

$$g(x_0) = \int \delta(x_0-\sum_{i=1}^{n} a_i x_i) \prod_{i=1}^{n} f(\alpha_i,\beta_i;x_i)\mathrm{d}x_i$$

where $\delta(x)$ is the Dirac delta function and

$$f(\alpha_i,\beta_i;x_i)=\frac{x_i^{\alpha_i - 1}(1-x_i)^{\beta_i - 1}}{\mathrm{B}(\alpha_i,\beta_i)}$$

for all $0<x<1$, and $f(\alpha_i,\beta_i;x_i)=0$ otherwise.

For example, it is known that the convolution of Gamma functions can be approximated by another Gamma function, whose mean and variance is the sum of the mean and variances of the convoluted Gamma distributions (1). Is there a similar result for the convolution of Beta distributed variables?

(1) Stewart, Trevor, et al. "A simple approximation to the convolution of gamma distributions." (2007).


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