Question: Standard random walk assumes a sequence of iid RVs $\{X\_i\}\_{i\geq 0}$ and studied the distribution of $S\_n=\sum\_{i=0}^n X\_i$. 

Here, I am wondering whether there is some work on 

$T\_n=\sum\_{i=0}^n \alpha^i X\_i$ where $\alpha\in (0,1)$ is a given fixed rational number. 

In particular, the properties of $\lim\_{n\rightarrow \infty} T\_n$, such as the distribution, etc. 

Any reference would be appreciated.

Answer: 
The special case when the $X\_i$'s are +1 or -1 with equal probabilities is called Bernoulli Convolution, see the nice survey by Peres, Schlag and Solomyak: [SIXTY YEARS OF BERNOULLI CONVOLUTIONS](http://math.uchicago.edu/~schlag/papers/sixty.pdf).
At some point I was looking for an answer to a particular case of your question (when $X\_i$ are iid uniform on [-1,1]) and did not succeed. There is some information in articles by [Borovkov](http://www.mathnet.ru/php/person.phtml?&personid=11308&option_lang=eng) published around 2008 but I would not say it gives an answer you probably expect to find.