Rao, Pathak and Koltchinskii have recently studied a sequential
approach to resampling in which resampling is carried out sequentially
one-by-one (with replacement each time) until the bootstrap sample contains $m
\approx (1 - e^{-1})n \approx 0.632n$ distinct observations from the original
sample. In our previous work, we have established that the main empirical
characteristics of the sequential bootstrap go through, in the sense of being
within a distance $O(n^{-3/4})$ from those of the usual bootstrap. However, the
theoretical justification of the second-order correctness of the sequential
bootstrap is somewhat difficult. It is the main topic of this investigation.
Among other things, we accomplish it by approximating our sequential scheme by
a resampling scheme based on the Poisson distribution with mean $\mu = 1$ and
censored at $X = 0$.