my problem is the following. I have a scalar variable, which changes with a certain noise every step:
Step 0: $x_0 = \nu_0$,
Step 1: $x_1 = s\cdot x_0 + \nu_1$,
...
Step i: $x_i = s\cdot x_{i-1} + \nu_i$,
where $s$ is positive scalar value, $\nu_i$ - zero-mean normally distributed gaussian RV (noise). Now, at certain step $k$, absolute value of $x_i$ exceeds a scalar threshold value $y$. Hence, $k$ is also a random variable. The question is, how to find the expected value of $k$ give the distribution of $\nu$, $s$ and $y$?
I'll appreciate any advice on how to approach the problem! For now I'm able to predict the distribution of absolute value of $x_i$ at any step $i$ (it is a half-normal distribution with parameters dependent on $\nu$ and $s$), but I'm totally failing to make a step towards finding the distribution of $k$.
You are summing Gaussians, so basically
$$x_0\sim N(0,\sigma^2)$$ $$x_1\sim N(0,\sigma^2+s\sigma^2)$$ $$x_2\sim N(0,\sigma^2+s\sigma^2+s^2\sigma^2)$$ And in general, using geometric series $$x_k\sim N(0,\sigma^2(\frac{s^k-1}{s-1}))$$