A strong law for a set-indexed partial sum process with applications to exchangeable and stationary sequences

AbstractConsider a set-indexed partial sum process {ΣjεJn Yj, n ⩾ 1} where {Yn, n ⩾ 1} are identically distributed random variables and {Jn, n ⩾ 1} is a nondecreasing sequence of finite sets of positive integers with j(n) = Card(Jn) → ∞. A strong law of the form ΣjεJn Yj/bj(n) → 0 almost certainly is established where {bn, n ⩾ 1} are constants with bn/n → ∞. As special cases, new results are obtained for exchangeable and stationary sequences. The result for stationary sequences strengthens a weak law proved by Maller [9] in that the convergence is shown to be almost certain