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>



The "Chaos Game" as a particular series of i.i.d. random variables
$\newcommand\al{\alpha}$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](https://en.wikipedia.org/wiki/Kolmogorov%27s_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](https://www.google.com/search?q=bernoulli%20convolution&oq=bernoulli%20convolution&aqs=chrome..69i57.7297j0j1&sourceid=chrome&ie=UTF-8). 


In the particular case when $X$ is $U(0,1)$ and $\alpha=1/2$, $F\_Y$ is the well-studied [Fabius function](https://www.google.com/search?q=fabius%20function&oq=fabius%20function&aqs=chrome..69i57j69i61.7180j0j1&sourceid=chrome&ie=UTF-8). 


