# A variant of random walk

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.

• I guess $\alpha^n$ should read $\alpha^i$. Sep 9, 2017 at 16:06

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.

• Among other things, for most given $\alpha$ it is unkown whether the law of $T_\infty$ is absolutely continuous. I find this mind-blowing. Sep 9, 2017 at 16:09

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)

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 published around 2008 but I would not say it gives an answer you probably expect to find.

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.