# geometric sum - weighted random walk

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)?

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You mean $\sum_{i=0}^n W_i\alpha^i$ I presume? – Siméon Jan 21 '13 at 10:35
thank you, edited. – Gianni Jan 21 '13 at 13:52

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.
Thank you very much for the answer and reference. In the identity you wrote do you intend $W$ is distributed as $W_0$ or as any of the samples of $W$ or did i miss something important about $W_1$? – Gianni Jan 21 '13 at 14:28