The "Chaos Game" as a particular series of i.i.d. random variables 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
 A: $\newcommand\al{\alpha}$Let us drop the factor $1-\al$, by considering 
$$Y:=X_\infty/(1-\al)=\sum_{k=0}^\infty\al^k X_k.$$
By Kolmogorov's three-series theorem, this series will converge almost surely (a.s.) unless at least one of the tails of the distribution of $X_0$ is too heavy. 
Assume that the series indeed converges a.s. 
Then, obviously, 
$$Y\overset D=X+\al Y,$$
where $\overset D=$ denotes the equality in distribution and $X$ is an independent copy of the $X_k$'s. So, we have the functional equation for $F_Y$: 
$$F_Y(y)=\int_{-\infty}^\infty F_Y((y-x)/\al)\,dF_X(x)$$
for real $y$, where $F_Z$ denotes the cdf of $Z$. Equivalently, we have the functional equation for $f_Y$:
$$f_Y(t)=f_Y(\al t)\,f_X(t)$$
for real $t$, where $f_Z$ denotes the characteristic function of $Z$. Of course, we can also write 
$$f_Y(t)=\prod_{k=0}^\infty f_X(\alpha^k t)$$
for real $t$. 
In the particular case when $X$ is Rademacher, the distribution of $Y$ is the well-studied Bernoulli convolution. 
In the particular case when $X$ is $U(0,1)$ and $\alpha=1/2$, $F_Y$ is the well-studied Fabius function. 
