geometric sum - weighted random walk

Question

I am trying to model the following sum:

$\sum_{i=0}^{n}{W_i \alpha^{i}}$

where $\alpha \in[0, 1) $ and $W_n$ takes values 0 or 1 and may be modeled as a markow chain or for simplicity as a binary random variable with $P(W=0)=p_0$ and $P(W=1)=1-p_0$

In the special case in which $n->\infty$ and $p_0=0$ the series should converge to $1\over{1-\alpha}$ and I suppose the infinite series will converge to $(1-p_0)\over{1-\alpha}$ when $p_0 \in (0,1]$.

I would like to understand whith what probability the sum above will be greater than a given threshold depending on the threshold itself, on the properties of $W_n$, on $n$ and of $\alpha$.

Is there some relevant theory I could use to model this problem (or a simplfied version of it)?

Answer 1

When $n\to\infty$, the random variables $X_n=\sum\limits_{i=0}^nW_i\alpha^i$ converge almost surely to some non deterministic limit $X=\sum\limits_{i=0}^{+\infty}W_i\alpha^i$. Obviously, $0\lt X\lt x_\alpha$ with $x_\alpha=\frac1{1-\alpha}$ with full probability. The distribution of $X$ is uniquely determined by the fact that the identity in distribution $$ W'+\alpha X'\stackrel{\text{dist.}}{=}X, $$ for some independent $W'$ and $X'$ distributed as $W$ in the question and as $X$, respectively. Not everything is known about the distribution of $X$, and by far, but the following results hold:

• In the special case $p_0=\frac12=\alpha$, the distribution of $X$ is uniform on $[0,1)$.
• When $p_0\ne\frac12=\alpha$, the distribution of $X$ is singular and without atom, on $[0,1)$.
• When $p_0=\frac12\gt\alpha$, the distribution of $X$ is singular and without atom, on $[0,x_\alpha)$.
• When $p_0=\frac12\lt\alpha$, to determine whether the distribution of $X$ is singular or not is called the Erdős problem.

To learn more about this exciting subject, see the survey Iterated random functions by Diaconis and Freedman, especially Section 2.5 for the model in this post.