  The prophet inequalities problem has received significant study over the past
decades and has several applications such as to online auctions. In this paper,
we study two variants of the i.i.d. prophet inequalities problem, namely the
windowed prophet inequalities problem and the batched prophet inequalities
problem. For the windowed prophet inequalities problem, we show that for window
size $o(n)$, the optimal competitive ratio is $\alpha \approx 0.745$, the same
as in the non-windowed case. In the case where the window size is $n/k$ for
some constant $k$, we show that $\alpha_k < WIN_{n/k} \le \alpha_k + o_k(1)$
where $WIN_{n/k}$ is the optimal competitive ratio for the window size $n/k$
prophet inequalities problem and $\alpha_k$ is the optimal competitive ratio
for the $k$ sample i.i.d. prophet inequalities problem. Finally, we prove an
equivalence between the batched prophet inequalities problem and the i.i.d.
prophet inequalities problem.
