# 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

$$\newcommand\al{\alpha}$$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.