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).
Well, $T\_n$ has the same distribution as $T\_n'$, defined by $T\_{n+1}':=\alpha T\_n' + X\_{n+1}$ with $T\_0':=X\_0$. The sequence $(T\_{n+1}')\_{n \ge 0}$ is an AR-1 sequence, and there is vast literature about it.