Random graphs and the strong convergence of bootstrap means (Q2703022)

This paper deals with graphs \(G_n\) which are generated by multi-sets \(I_n\) having \(n\) random integers as elements such that the vertices of \(G_n\) are connected by edges if the elements of \(I_n\) which the vertices represent are the same. The authors present asymptotic results on the sparsity of edges connecting the different subgraphs \(G_n\) of the random graph generated by \(\bigcup^\infty_{n=1} I_n\). Two models of the bootstrap are considered---the first one is a triangular array with independent rows, and the second one is induced by a single sequence \(U_1,U_2,\dots\) of independent and over \((0,1)\) uniformly distributed random variables (here the rows are not independent). The authors use the results obtained here to link almost sure and complete convergence of the corresponding bootstrap means and the averages of randomly chosen subsequences of a sequence of random variables which are independent, identically distributed, and have a finite mean. Complete convergence of these means and averages is given in terms of a relationship between a moment condition on the bootstrap sample size and the bootstrapped sequence. Some results on bootstrap means published earlier have been corrected in this paper, and the paper concludes by stating some conjectures for future research.