Suppose we have a series of RVs with $X_i \sim D$. In our instance, let's set $D$ to be Bernoulli, i.e. $X_i \sim Bern(p)$.
For some $W > 0$, we can now also create a series of RVs $Y_i = \sum_{k = 1}^W X_{i - k + 1}$, i.e. where $Y_i$ looks at a sliding window of size $W$ of the $X_i$ series.
What techniques can we use to study $Y_i$? A classic question is to ask: what is the probability that $Y_i \geq T$ at any $i \leq t$ for some time $t$; this problem is challenging because $Y_i, Y_{i - 1}, $ etc. will often not be independent. For example, how might we produce upper and lower bounds on: $$ \mathbb{P}\left[\bigcup Y_i \geq T \right] $$
Some prior questions that were not answered fully include: Bernoulli trials required for k successes within a sliding window, Expected number of tosses under a moving-window-based stopping condition
