A curious application of the Borel-Cantelli Lemmas, a result of Barndorff-Nielsen, and some open problems. Issue 3 (4th May 2019)

• Record Type: Journal Article
• Authors: Chandra, Tapas Kumar; Rosalsky, Andrew
• Is Part Of: Stochastic analysis and applications. Volume 37:Issue 3(2019)
• Page Start: 473
• Page End: 479
• Publication Date: 2019-05-04
• Subjects: First Borel-Cantelli Lemma -- Second Borel-Cantelli Lemma -- Kolmogorov 0-1 law; 60A99 -- 60F20; Stochastic analysis -- Periodicals; 519.2205
• DOI: 10.1080/07362994.2019.1585262
• Languages: English
• ISSNs: 0736-2994

Abstract

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.