PM5M Memory-Dependent Limit Theorems: Resolving the Long-Standing Problem of Weighted Borel–Cantelli for Dependent Events

DescriptionThis work presents a decisive advance in the theory of dependent events in probability, focusing on the weighted Borel–Cantelli framework under memory effects. The results are obtained in the PM5M (Probabilistic Modal 5-Memory) formal system — a newly introduced probabilistic–modal logic with a dedicated memory operator. For decades, the generalization of the Borel–Cantelli lemmas to dependent events with long-range memory has been a notoriously difficult problem, with partial results available only under restrictive assumptions (independence, strong mixing, or fast covariance decay). Our approach resolves this gap by proving sharp limit theorems for processes with memory, using a combination of: Weighted Kochen–Stone inequalities for dependent events; Covariance decay conditions in the sense of Peligrad; Memory-weight divergence criteria unifying trial frequency and historical dependence; A complete measure-theoretic foundation (Kolmogorov extension, Carathéodory outer measure, monotone-class transfer). The paper establishes necessary and sufficient divergence criteria for infinitely-often occurrence of events in dependent-memory processes, and provides a constructive counterexample showing the sharpness of these criteria. Beyond its theoretical importance, the result opens the way for applications in: Statistical mechanics with long-range correlations; Reliability theory under persistent dependencies; Machine learning with recurrent feedback loops; Network theory with memory-driven connectivity patterns. This work not only extends a classical pillar of probability theory but also introduces PM5M as a formal language capable of expressing and proving advanced probabilistic statements involving memory — a capacity previously unreachable in standard logical frameworks.