The asymptotic distribution of the bootstrap sample mean of an infinitesimal array
The paper offers asymptotic results for the bootstrap of the sample mean given by an infinitesimal array of rowwise i.e.d. random variables with convergent partial sums. It is known that the ordinary bootstrap works if the central limit theorem holds for the underlying array, see for instance \textit{E. Mammen} [When does bootstrap work? Asymptotic results and simulations. (1992; Zbl 0760.62038)]. If the limit law has a nontrivial Poisson part then the bootstrap statistic may have a data-dependent limit law. These laws are typically infinitely divisible. Results of this type are also discussed when the bootstrap sample size \(m_n\) and the original sample size \(k_n\) of the data fulfill \(m_0= o(k_n)\) or \(k_n= o(m_n)\) and \(\min (k_n,m_n) \to \infty\). The results and the methods of proof are very interesting.
scientific article; zbMATH DE number 1156742