# Solve non-homogeneous recurrence relation

I'm stuck on a recurrence relation that arises in a simulation I'm writing. Does anybody know how to proceed on this? I'm not even sure, because of the variable coefficient, how to get the associated homogeneous case.

$$w_t = w_{t-1} \epsilon (1-u_t) + \epsilon u_t$$

$\epsilon$ is a constant and ${u_t}$ is a binary sequence

Thanks

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What's a binary sequence? If it is what I'd think it would be, a sequence of $0$s and $1$s, why is $u_t$ being squared? – joriki Mar 20 '12 at 15:24
@joriki: In the simplest case, it is although in subsequent elaborations it might not be. In any case, it's squared because this equation is derived from two others. Should I put the derivation up? (I thought it would make the question less general.) – mac389 Mar 20 '12 at 18:50
It's not very helpful to say that $u_t$ is a binary sequence and then not say what this might mean "in subsequent elaborations"; then you're not actually saying anything at all when you say that the sequence is binary. You should decide whether you want $u_t$ to be just any sequence of real (or complex?) numbers or whether you want to place restrictions on $u_t$. – joriki Mar 20 '12 at 18:59
@joriki: My apologies. Please consider $u_t$ a binary variable. I'll edit my post to drop the exponent. – mac389 Mar 20 '12 at 19:01
What exactly do you mean by "solve"? Are you looking for a "closed" form? The limit of $w_t$ as $t \to \infty$? What? – Aryabhata Mar 20 '12 at 19:05

## 1 Answer

Whenever $u_t=1$, we have $w_t=\epsilon$, and whenever $u_t=0$, we have $w_t=w_{t-1}\epsilon$. Thus the sequence consists of initial segments of the geometric sequence $\epsilon^n$ and restarts at $\epsilon$ whenever $u_t=1$.

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Thank you for your help. Is there a closed form for this if one assumes that $u_t$ has a certain average number of 1's and 0's? – mac389 Mar 20 '12 at 19:45
@mac389: I don't see how that could be, since the values will depend on the positions of the $1$s and $0$s. I don't think there's any more to say about the form of $w_t$ than what I said, though of course you can look at statistical properties of $w_t$ in dependence on statistical properties of $u_t$. – joriki Mar 20 '12 at 19:53
Thanks, I will do just that. – mac389 Mar 20 '12 at 20:01