Fix a parameter $\alpha\in(0,1)$ and take an i.i.d. sequence $X_0,X_1,\ldots$ of $\mathbb{R}^n$ valued random variables. Construct the limiting random variable

$X_\infty = (1-\alpha)\sum_{k=0}^\infty \alpha^k X_k.$

Is any general result known about this kind of limit? What if the $X_i$ follow a well known distribution like uniform/Rademacher?

I was motivated by this sum after running into: https://en.wikipedia.org/wiki/Chaos_game. For example if the $X_i$ are uniformly distributed on the 3 vertices of a triangle and $\alpha = 1/2$ the limiting distribution is supported on the associated Sierpinski Triangle. The fact that finite support distributions can give fractal shapes from this construction leads me to believe this is a non-trivial question.

My apologies if this ends up being an exercise in some well known textbook on probability theory. If it is, I'd appreciate a reference for that textbook.

Edit: I was able to locate http://u.math.biu.ac.il/~solomyb/RESEARCH/Bernotes.pdf