A curious application of the Borel-Cantelli Lemmas, a result of Barndorff-Nielsen, and some open problems. Issue 3 (4th May 2019) - Chandra, Tapas Kumar; Rosalsky, Andrew
Abstract: For an arbitrary sequence of events { A n, n ≥ 1 } and an integer m ≥ 1, we study the problem whether ∑ n = 1 ∞ P ( A n ) P ( A n + m c ) < ∞ implies either ∑ n = 1 ∞ P ( A n ) < ∞ or ∑ n = 1 ∞ P ( A n c ) < ∞ . It is partially solved using both probabilistic and nonprobabilistic arguments. The First and Second Borel-Cantelli Lemmas are both used to show that the implication holds if m  = 1 and an equivalent formulation concerning a real series problem is presented. That the implication does not hold if m  = 2 is shown using a real series and it is an open problem for m ≥ 3. Some other open problems are also presented and discussed.