Second-order correctness of the Poisson bootstrap

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 ≈ (1 - e ^-1)n ≈ 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 μ = 1 and censored at X = 0.