(Cross post from MSE)
Let $\xi_i$ be iid random variables, and define:
$$S_{(k)} = \sum_{i=1}^N \xi_{i+k}$$
Now, define:
$$\tau = \min \left\{ k : S_{(k)} \notin (a,b) \right\}$$
How can I find $\mathbb{E}[\tau]$ or even $P(\tau < t)$, and whether there are any conditions under which these do not depend on the specific distribution of $\xi_i$ but on its first two moments?
To be more precise, I would expect to be able to express $\mathbb{E}[\tau]$ in terms of $N$, the statistical properties of $S_{(0)}$ (for instance, in case we knew the initial state of the register and would be interested in larger values of $k$) and the moments of $\xi_i$. This is usually the case for problems in which one new iid is added to the sum when increasing $k$.
However, here $S_{(k)}$ acts as a shift register/buffer/sliding window/etc., where the oldest value is discarded every time a new one is added. Moreover, other techniques used in similar situations, like Wald's Identities, cannot be applied here, due to the sum taking place over a fixed number $N$ of iids.
Indeed, it turns out that one can prove that the following is a martingale:
$$M_n=S_n\left(\frac{N}{N-1} \right)^n - \mu \sum_{i=1}^n \left(\frac{N}{N-1} \right)^i$$ but the exponential terms prevent us from using the OST since $M_n$ is not uniformly integrable.
Finally, I find it strange that there seems not to be a lot of research about the statistics of these moving windows/shift registers, but I am not able to find any bibliography on this type of problems.
