Let $Z_t = \psi_t |\lambda Z_{(t-1)} + (1-\lambda)\epsilon_t |$ be a random variable where $\epsilon~N(0,1)$ is a Gaussian distributed number, $Z_0 = z_0$ and $\psi \in [-1,1]$ a random variable, either -1 or + 1. The parameter $0>\lambda>1$ decays the seed value $z_0$ and hence it lies between 0 and 1.
Inside the Abs, we have a Gaussian distributed number (at t = 1, anyway). The absolute of this number, multiplied by $\psi$, is then on either side of zero with equal probability. The p.d.f. will thus look something like this:
$f_Z(x) = \frac{1}{2}(\frac{\exp(-\frac{(x-\mu)^2)}{2\sigma^2})}{\sigma \sqrt{2\pi}})+\frac{1}{2}(\frac{\exp(-\frac{(x+\mu)^2)}{2\sigma^2})}{\sigma \sqrt{2\pi}}),$
where $\mu$ and $\sigma$ are the mean and standard deviation of the original random number inside the absolute value. Suppose we call this distribution a $Z(\mu, \sigma)$ distribution. It is easy to see that when $t>1$, the variable $Z_t$ will be distributed $Z(\mu,\sigma)$, and also (I think) the entire contents inside the absolute brackets. After a bit of analysis, we also see that $\mu := \lambda^t z_0$ and $\sigma := \sqrt{\frac{(\lambda-1)(\lambda^{2t}-1)}{\lambda+1}}$. Another important observation is that the $\psi$ random number scrambles the time series and remove any autocorrelation in the temporal dimension. With this background, here come the question!
1) Are $Z_t$ and $Z_{(t+1)}$ independent?
2) What is the distribution of $\sum_t Z_t$? Is it simply the convolutions of $Z_1,Z_2,...,Z_(t-1),Z_t$?