A zero-one law approach to the central limit theorem for the weighted bootstrap mean
Let \(X_j\), \(j \geq 1\), be a sequence of independent and identically distributed random variables with finite variance, and let \(w_n = (w_n (1), \dots, w_n (n))\), \(n \geq 1\), be a sequence of independent vectors of random weights, independent of the data sequence and such that the components \(w_n (j) \geq 0\) of \(w_n\) are exchangeable for every \(n \geq 1\) and satisfy \(\sum^n_{j = 1} w_n (j) = 1\). Then the weighted bootstrap sample mean for sample size \(n\) is defined by \(\overline X^*_n = \sum^n_{j = 1} w_n (j) X_j\). \textit{D. M. Mason} and \textit{M. A. Newton} [Ann. Stat. 20, No. 3, 1611-1624 (1992; Zbl 0777.62045)] were the first to prove a conditional central limit theorem for \(\overline X^*_n\) given the data, under appropriate further conditions on the weights. Their proof is based on an additional randomness which is brought into play via the introduction of ranks that are independent of the data and the weights. This device makes Hajek's central limit theorem for linear rank statistics applicable. In the present paper a new technique is developed to obtain the result. Through zero-one laws it is shown that this conditional central limit theorem can be derived from an unconditional one which itself follows from the standard Lindeberg central limit theorem via a conditioning with respect to the weights. The applicability of the technique to independent, not necessarily identically distributed random variables \(X_j\) is also pointed out.
scientific article; zbMATH DE number 926227