Expected value of a discrete-time process

Question

I have a discrete-time process

$$Z(t+1) = X(t)Z(t) + Y(t)$$

where $X$ and $Y$ are i.i.d random variables with known distributions. All variables are real. Is it possible to compute the expected value

$$\lim_{t \rightarrow \infty} \langle Z(t)\rangle$$

You may assume that at time $t=0$ the value of $Z = Z_0$ is known. In my particular problem, $X\sim U(0,1)$ and $Y \sim \mathrm{Bernoulli}(p)$, but a general solution is prefered if possible.

Answer 1

$Z_t$, being a function of $Z_{t-1},$ $X_{t-1}$ and $Y_{t-1},$ is independent of $X_t$. Hence, $\mathbb{E}Z_t=\mathbb{E}Z_{t-1}\mathbb{E}X_{t-1}+\mathbb{E}Y_{t-1}$. In your case, the first factor is $0$, so it all equals the mean of the $Y$ variables, no matter $t$.

In case $\mathbb{E}X_t\neq 0,$ it looks like you get $\mathbb{E}Z_t=Z_0(\mathbb{E}X_0)^{t}+\sum_{n=0}^{t-1}(\mathbb{E}X_0)^n \mathbb{E} Y_0$ by induction.

Now, this is just a geometric series, which you should be able to treat.