A curious application of the Borel-Cantelli Lemmas, a result of Barndorff-Nielsen, and some open problems

Abstract

For an arbitrary sequence of events $\{A_n,n\ge 1\}$ and an integer $m\ge 1,$ we study the problem whether ${\sum }_{n=1}^{\infty }P(A_n)P(A_{n+m}^c)<\infty$ implies either ${\sum }_{n=1}^{\infty }P(A_n)<\infty$ or ${\sum }_{n=1}^{\infty }P(A_n^c)<\infty .$ 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 $m\ge 3.$ Some other open problems are also presented and discussed.

Keywords:

• First Borel-Cantelli Lemma
• Second Borel-Cantelli Lemma
• Kolmogorov 0-1 law

Mathematics Subject Classification:

• 60A99
• 60F20