A central limit theorem for bootstrap sample sums from non-i.i.d. models

•For bootstrap sample sums a very general central limit theorem is established.•The parent sequence does not need to be independent or identically distributed.•The main result extends a result of Liu (1988).•A version of the main result is also obtained with random bootstrap sample size.

For bootstrap sample sums resulting from a sequence of random variables {Xn,n≥1}, a very general central limit theorem is established. The random variables {Xn,n≥1} do not need to be independent or identically distributed or to be of any particular dependence structure. Furthermore, no conditions, including moment conditions, are imposed in general on the marginal distributions of the {Xn,n≥1}. As a special case of the main result, a result of Liu (1988) concerning independent but non-identically distributed {Xn,n≥1} is extended to a larger class of parent sequences. A version of the main result is also presented wherein the bootstrap sample sums have random bootstrap sample sizes.