A strong law for a set-indexed partial sum process with applications to exchangeable and stationary sequences
Let \(\{Y_ n\), \(n\geq 1\}\) be a sequence of identically distributed random variables, not necessarily independent nor with finite mean, and let \(\{J_ n\), \(n\geq 1\}\) be an increasing sequence of finite sets of positive integers. Set \(S_ n=\sum_{j\in J_ n}Y_ j\), and let \(\{\) b(n), \(n\geq 1\}\) be any sequence of constants such that \[ n^{- 1}b(n)\to \infty \quad and\quad \sum_{n\geq 1}P[| Y_ 1| >b(n)]<\infty. \] Then, provided an ergodic theorem holds for the partial sums of \(g(| Y_ j|)\), for any \(g\in L_ 1\), \(S_ n/b(| J_ n|)\to 0\) a.s. The result is applied to obtain laws of large numbers for symmetric functions of exchangeable sequences, for stationary sequences and for sequences of pairwise independent random variables.
sequences of pairwise independent random variables
scientific article; zbMATH DE number 4032704
scientific article