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).
In the physics literature, these are known as random walks with shrinking steps. The distribution for $\alpha = {\frac{1}{2}}$, the golden ratio, and $\alpha = 2 ^ {\frac{1}{n}}$, where n is a positive integer and the $X\_i$'s are +1 or -1, are discussed in Krapivsky and Redner's American Journal of Physics article. Some possibly useful references are: 

Serino, C.A. and Redner, S.,"The Pearson walk with shrinking steps in two dimensions",Journal of Statistical Mechanics: Theory and Experiment, volume = 1,01006 (2010)

Rador, T.,"Random walkers with shrinking steps in d dimensions and their long term memory",Physical Review E,74,051105,(2006)

Rador, T. and Taneri, S.,"Random walks with shrinking steps: First-passage characteristics", Physical Review E 73, 036118 (2006)

Krapivsky, P.L. and Redner, S.,"Random walk with shrinking steps",American Journal of Physics 72,591-598 (2004)