A limit theorem for expectations conditional on a sum
Let \((\xi_ n)\) be i.i.d. and let \(U\) be another r.v. such that \(U\) and \(\xi_ 1\) may be dependent, but \(U\) is otherwise independent of the sequence. The main result of the paper is that, under appropriate regularity and third moment conditions, \[ \mathbb{E}[U \mid \xi_ 1 + \cdots+\xi_ n=n \mu]=\mathbb{E}[U]+O(n^{-1}), \] and this result is the best possible in the sense that the rate of convergence can be of strictly lesser order if a third moment does not exist. The primary tool employed in the proof of the above equation is a formula for the conditional expectation due to Bartlett; cf. \textit{S. Zabell} [Ann. Probab. 7, 159-165 (1979; Zbl 0392.60013)].
scientific article; zbMATH DE number 179064