On the law of large numbers for the bootstrap mean (Q1198986)

Let \(X_ 1,X_ 2,\dots\) be an infinite sequence of independent and identically distributed random variables. For each \(n=1,2,\dots\), let \(X^{(n)}_ 1,\dots,X^{(n)}_ m\) be the ordinary Efron bootstrap sample from \(X_ 1,\dots,X_ n\) with bootstrap sample size \(m=m(n)\). The author proves, among others, that \[ \lim_{n\to\infty} m^{- 1}\sum^ m_{j=1} X^{(n)}_ j=E(X)\qquad\text{almost surely} \] if for some \(\delta\) \((0\leq\delta<1)\) \[ E(| X|^{1+\delta})<\infty\qquad\text{and}\qquad\lim_{n\to\infty} n^{1-\delta}/m<\infty, \] or if for some \(r=1,2,\dots\) and \(\delta>0\), \[ E(X^ 2)<\infty\qquad\text{and}\qquad\limsup(\log n \log_ 2\cdots\log_{r-1}n (\log_ r n)^{1+\delta})/m(n)<\infty, \] where \(\log_ r\) stands for the \(r\)-times iterated logarithm. The results improve on those of \textit{K. Athreya} [ibid. 1, 147-150 (1983; Zbl 0518.62016)].